Nivolumab or Expectant Observation Following Ipilimumab, Nivolumab, and Surgery in Treating Patients With High Risk Localized, Locoregionally Advanced, or Recurrent Mucosal Melanoma

Inclusion Criteria: * STEP 1 ELIGIBILITY CRITERIA * Documentation of disease: * Histologic documentation: histologically proven mucosal melanoma by local pathology * Tumor tissue: tumor tissue from the primary site of disease must be available for PD-L1 testing (stratification factor) * Disease status * Tumors must have NOT been completely resected, or must be locoregionally recurrent if previously resected; tumor must be deemed potentially resectable by local surgeon * MM arising from the head/neck, genitourinary, or gastrointestinal tract * Disease meets any 1 of 4 characteristics: * Regional lymph node (LN) involvement; OR * Multifocal/satellite primary disease; OR * Single localized, primary disease meeting one of the following site-specific requirements: * Head/neck - any primary lesion if sinonasal; pT4a or above for nasal or oral cavity * Anorectal - any primary lesion * Conjunctiva - any primary lesion T2 or T3 stage by American Joint Committee on Cancer (AJCC) * Vaginal/cervical - any primary * Vulvar (hair bearing surface, labia majora) - AJCC cutaneous stage IIB or higher * Esophageal - any primary * Locoregionally recurrent following prior resection * No evidence of metastatic disease at the time of registration * No prior medical therapy (chemotherapy, immunotherapy, biologic or targeted therapy) or radiation therapy for MM, unless locoregionally recurrent; if recurrent, no prior medical or radiation therapy is allowed for the latest recurrence * No history of the following: * Active known or suspected autoimmune disease * Human immunodeficiency virus (HIV) with CD4+ count \< 300 or detectable viral load; patients with HIV, undetectable viral load, and CD4+ count \>= 300 are eligible * Known active hepatitis B or C * Hepatitis B can be defined as: * Hepatitis B virus surface antigen (HBsAg) \> 6 months * Serum hepatitis B virus (HBV) deoxyribonucleic acid (DNA) 20,000 IU/ml (105 copies/ml), lower values 2,000-20,000 IU/ml (104-105 copies/ml) are often seen in hepatitis B virus e antigen (HBeAg)-negative chronic hepatitis B * Persistent or intermittent elevation in alanine aminotransferase (ALT)/aspartate aminotransferase (AST) levels * Liver biopsy showing chronic hepatitis with moderate or severe necroinflammation * Hepatitis C can be defined as: * Hepatitis C antibody (AB) positive * Presence of hepatitis C virus (HCV) RNA * Known active pulmonary disease with hypoxia defined as oxygen saturation \< 85% on room air * Not pregnant and not nursing * For women of childbearing potential only, a negative pregnancy test done =\< 14 days prior to registration is required * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Absolute neutrophil count (ANC) \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Creatinine clearance \>= 30 mL/min by Modified Diet in Renal Disease (MDRD) equation or Cockcroft-Gault * Total bilirubin =\< 1.5 x upper limit of normal (ULN) * Except in case of Gilbert disease * AST/ALT =\< 2.5 x upper limit of normal (ULN) * Thyroid-stimulating hormone (TSH) within normal limits (WNL) * Supplementation is acceptable to achieve a TSH WNL; in patients with abnormal TSH if free T4 is normal and patient is clinically euthyroid, patient is eligible * Concomitant medications * No systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 7 days of registration (inhaled steroids for patients with underlying chronic pulmonary disease is acceptable as long as they meet other eligibility as listed above) * No other planned concurrent investigational agents or other tumor directed therapy (chemotherapy, radiation) while on study * STEP 2 ELIGIBILITY CRITERIA * Surgical resection of all gross disease * This assessment will be made by the local investigator based on review of the operative report, pathology results, and/or radiology reports; microscopically positive margins (e.g. R1 resection) are permitted * Completion of PD-L1 testing * Patients will be stratified as PD-L1 \>= 5% versus (vs) \< 5% OR inevaluable; baseline tumor will be utilized; if this returns inevaluable, efforts should be made to utilize the resected specimen * Randomization within 112 days of completion of surgical * The primary region must be included on cross-sectional imaging (e.g. sinus/neck if arising from sinonasal primary; pelvis if genitourinary); radiographic changes considered nonspecific or possibly due to surgery or radiation are not considered evidence of disease * No significant immune-related adverse event due to the nivolumab plus ipilimumab dose received in the neoadjuvant setting, as follows: * Any grade myocarditis, including an asymptomatic troponin elevation felt to be related to nivolumab plus ipilimumab * Any grade neurologic complication (e.g. meningitis, encephalitis, neuropathy); study chair should be contacted if there is any question whether an adverse event (AE) was neurologic in nature * Grade 2 or higher pneumonitis * Grade 2 colitis * Grade 2 or higher anemia (due to drug; unrelated anemia is not exclusionary) * Thyroid/adrenal AEs regardless of grade that have stabilized or are clinically asymptomatic are eligible * Fatigue, regardless of grade, is not a contraindication to randomization * Grade 4 AST or ALT elevation * Asymptomatic elevated amylase and lipase, regardless of grade, are not contraindications to randomization * Grade 4 rash; grade 3 rash is not a contraindication to randomization; any oropharyngeal lesions or bullous skin lesions that are suggestive of toxic epidermal necrolysis or Stevens-Johnson syndrome are contraindications to randomization regardless of the grade of rash * If not specified above, other Common Terminology Criteria for Adverse Events (CTCAE) grade 3 or higher AEs deemed possibly related to the nivolumab plus ipilimumab are exclusions to randomization; AEs that were attributable to surgery or adjuvant RT would not be contraindications to randomization