A Study to Determine Whether Chemotherapy and Atezolizumab is Better Than Chemotherapy, Bevacizumab and Atezolizumab in Patients With Advanced Liver Cancer

Inclusion Criteria: * Patient must be \>= 18 years of age * Patient must have a histologically confirmed diagnosis of combined hepatocellular carcinoma-cholangiocarcinoma (cHCC-CC) at the local laboratory based on the 2019 World Health Organization (WHO) classification, including the classical type and intermediate cell carcinoma * The classical type defines primary liver carcinoma with unequivocal features of both HCC and CC differentiation within the same tumors on routine histopathology with hematoxylin and eosin stains regardless of the proportion of each histology observed * The intermediate cell carcinoma defines cancers with biphenotypic differentiation in which cells have a morphology intermediate between hepatocytes and cholangiocytes. Intermediate cell carcinoma may be associated with expression of both hepatocyte and cholangiocytic markers. Distinct HCC and CC arising in the same liver, fibrolamellar HCC, morphologically typical HCCs with only immunohistochemical expression of keratin or other cholangiocytic markers, or morphologically typical CCs with only immunohistochemical expression of hepatocytic markers will be excluded * NOTE: Local pathology review constitutes adequate documentation of histology for initial study enrollment and treatment * Patient must have Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 * Patient must have disease which is unresectable or metastatic * Patient must not have any prior history of systemic therapy for advanced cHCC-CC. Prior adjuvant treatment composed of chemotherapy agents such as capecitabine or gemcitabine-based treatments are allowed if adjuvant treatment if at least 6 months have elapsed since completing chemotherapy at the time of enrollment * Patient must be Child Pugh class A * Patients with prior locoregional therapy are eligible provided the following are met: * Prior loco-regional therapy including surgical resection, chemoembolization, radiotherapy, or ablation was completed \> 4 weeks prior to randomization * Treated target lesion has increased in size by \> 25% or the target lesion was not treated with loco-regional therapy * Patients treated with palliative radiotherapy for symptoms must have completed radiotherapy \> 7 days prior to randomization and the target lesion must not have been the treated lesion * Patient must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used. * All patients of childbearing potential must have a blood test or urine study within 14 days prior to randomization to rule out pregnancy * A patient of childbearing potential is defined as anyone, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Patient must not expect to conceive or father children by abstaining from sexual intercourse or by using accepted and effective method(s) of contraception while on protocol treatment and for 6 months after the last dose of protocol treatment. Accepted and effective method(s) of contraception include those with a failure rate of \< 1% per year including bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovluation, hormonal releasing intrauterine devices, and copper intrauterine devices. Periodic abstinence (e.g. calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not adequate methods of contraception * Patient must have the ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver and/or family member available will also be considered eligible * Leukocytes \>= 3,000/mcL (must be obtained =\< 14 days prior to protocol randomization) * Absolute neutrophil count (ANC) \>= 1,500/mcL (must be obtained =\< 14 days prior to protocol randomization) * Hemoglobin \>= 9 g/dL (Patient may be transfused to meet this criterion) (must be obtained =\< 14 days prior to protocol randomization) * Platelets \>= 80,000/mcL (must be obtained =\< 14 days prior to protocol randomization) * Total bilirubin =\< 5 x institutional upper limit of normal (ULN) (must be obtained =\< 14 days prior to protocol randomization) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 5.0 x institutional ULN (must be obtained =\< 14 days prior to protocol randomization) * Creatinine =\< 1.5 x institutional ULN (must be obtained =\< 14 days prior to protocol randomization) * International normalized ratio (INR) =\< 1.5 x Institutional ULN (for patients not receiving anticoagulant therapy) (must be obtained =\< 14 days prior to protocol randomization). For patients receiving therapeutic anticoagulation, the patient must be on a stable anticoagulant regimen * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months of randomization are eligible for this trial * For patients with evidence of prior or active hepatitis B virus (HBV) infection (positive hepatitis B surface antigen \[HBsAg\] test and/or positive total hepatitis B virus core antibody \[HBcAb\] test at screening), the patient must be on suppressive therapy, for at least 2 weeks prior to randomization and willing to continue antiviral treatment for the length of the study * Patient must not have new or progressive brain metastases (active brain metastases) or leptomeningeal disease * Patients must not have laboratory evidence of active co-infection of HBV (positive HBsAg test) and hepatitis C virus (HCV) (detectable HCV ribonucleic acid \[RNA\]). Patients with a history of HCV infection but who are negative for HCV RNA by polymerase chain reaction (PCR) will be considered non-infected with HCV * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better * Patient must have measurable disease. Baseline measurements and evaluations of all sites of disease must be obtained within 4 weeks prior to randomization * Patients with untreated or incompletely treated varices with bleeding or high-risk for bleeding must undergo an esophagogastroduodenoscopy (EGD), and all size of varices (small to large) must be assessed and treated per local standard of care prior to randomization. Patients who have undergone an EGD within 6 months of prior to randomization do not need to repeat the procedure * Patient must not have had a prior allogenic bone marrow or solid organ transplant * Patient must not have a history of idiopathic pulmonary fibrosis, organizing pneumonia, drug-induced pneumonitis, or idiopathic pneumonitis, or evidence of active pneumonitis on baseline chest computed tomography scan * Patient must not have active or a history of autoimmune disease or immune deficiency, including, but not limited to, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjogren syndrome, Guillain-Barre syndrome, or multiple sclerosis, with the following exceptions: * Patients with a history of autoimmune-related hypothyroidism who are on thyroid-replacement hormone are eligible * Patients with controlled type 1 diabetes mellitus who are on an insulin regimen are eligible for the study * Patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis are excluded) are eligible provided all of following conditions are met: * Rash must cover \< 10% of body surface area * Disease is well controlled at baseline and requires only low-potency topical corticosteroids * There is no occurrence of acute exacerbations of the underlying condition requiring psoralen plus ultraviolet A radiation, methotrexate, retinoids, biologic agents, oral calcineurin inhibitors, or high-potency or oral corticosteroids * Patient must not have received prior treatment with immune checkpoint blockade therapies, including anti-CTLA-4, anti-PD-1, and anti-PD-L1 therapeutic antibodies * Patient must not be on treatment with systemic immunosuppressive medication (including, but not limited to, corticosteroids, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-TNF-alpha agents) within 2 weeks prior to randomization, or anticipate the need for systemic immunosuppressive medication during study treatment, with the following exceptions: * Patients who received acute, low-dose systemic immunosuppressant medication or a one-time pulse dose of systemic immunosuppressant medication (e.g., 48 hours of corticosteroids for a contrast allergy) are eligible * Patients who received mineralocorticoids (e.g., fludrocortisone), corticosteroids for chronic obstructive pulmonary disease (COPD) or asthma, or low-dose corticosteroids for orthostatic hypotension or adrenal insufficiency are eligible * Patient must not have inadequately controlled arterial hypertension (defined as systolic blood pressure (BP) \>= 150 mmHg and/or diastolic blood pressure \> 100 mmHg) prior to randomization. Patients may be on antihypersensitive medications to meet and maintain this criteria * Patient must not have significant vascular disease (e.g., aortic aneurysm requiring surgical repair or recent peripheral arterial thrombosis) within 6 months prior to randomization * Patient may not have a history of abdominal or tracheoesophageal fistula, gastrointestinal (GI) perforation, or intra-abdominal abscess within 6 months prior to randomization * Patient must not have any evidence of bleeding diathesis or significant coagulopathy (in the absence of therapeutic anticoagulation) * Patient must not have uncontrolled tumor-related pain. Patients requiring pain medication must be on a stable regimen at the time of randomization * For patients with symptomatic lesions (e.g., bone metastases or metastases causing nerve impingement) amenable to palliative radiotherapy should be treated prior to randomization. Patients must be recovered from the effects of radiation prior to randomization. There is no required minimum recovery period * For patients with asymptomatic metastatic lesions that would likely cause functional deficits or intractable pain with further growth (e.g., epidural metastasis that is not currently associated with spinal cord compression) they must be considered for loco-regional therapy if appropriate prior to randomization * Patient must not have uncontrolled pleural effusion, pericardial effusion, or ascites requiring recurrent drainage procedures (once monthly or more frequently). Patients with indwelling catheters (e.g., PleurX) are allowed * Patient must not have active tuberculosis * Patient must not have undergone any major surgical procedure, other than for diagnosis, within 4 weeks prior to randomization, or have the anticipation of need for a major surgical procedure during the study * Patient must not have any other disease, metabolic dysfunction, physical examination finding, or clinical laboratory finding that contraindicates the use of the agents used on this study, may affect the interpretation of the results, or may render the patient at high risk from treatment complications * Patient must not have received any live, attenuated vaccines (e.g., FluMist \[registered trademark\]) within 4 weeks prior to randomization, during treatment with atezolizumab, and for 5 months after the last dose of atezolizumab * Patient must not have received any treatment with investigational therapy within 28 days prior to randomization * Patient must have not received treatment with systemic immunostimulatory agents (including, but not limited to, interferon and interleukin 2 \[IL-2\]) within 4 weeks or 5 half-lives of the drug (whichever is longer) prior to randomization * Patients must not have a history of severe allergic anaphylactic reactions to chimeric or humanized antibodies or fusion proteins * Patient must not have a known hypersensitivity to Chinese hamster ovary cell products or to any component of the atezolizumab formulation * Patient must not have a known allergy or hypersensitivity to any component of the atezolizumab and bevacizumab formulation