Evaluation of Alternative Site Goserelin Acetate Injection for Ovarian Function Suppression (OFS) in Local and Locally Advanced Premenopausal Hormone Receptor Positive Breast Cancer Patients

Inclusion Criteria: * REGISTRATION (STEP 1): Age ≥ 18 years and ˂ 50 years * REGISTRATION (STEP 1): Have histologically or cytologically confirmed, localized or locally advanced hormone positive breast cancer stage I-III (defined as ER Immunohistochemistry (IHC) \> 1%\] having completed curative intent therapy and clinically in remission * REGISTRATION (STEP 1): Currently receiving ovarian function suppression (OFS) with use of goserelin on a monthly basis in the abdomen and either aromatase inhibitor or tamoxifen for at least 6 months prior to study enrollment for treatment of hormone receptor positive breast cancer with plan to continue medical OFS for at least the next 12 months * REGISTRATION (STEP 1): Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * REGISTRATION (STEP 1): Provide written informed consent * REGISTRATION (STEP 1): Ability to complete questionnaire(s) by themselves or with assistance * REGISTRATION (STEP 1): Willingness to provide mandatory blood specimens for correlative research * REGISTRATION (STEP 1): Willing to return to enrolling institution for follow-up (during the active monitoring phase of the study) * REGISTRATION (STEP 1): Negative serum pregnancy test =\< 14 days prior to registration, and a negative urine pregnancy test =\< 7 days prior to randomization * REGISTRATION (STEP 1): Sexually active patients and their partners must use an effective method of contraception associated with a low failure rate prior to study entry and for the duration of study participation and for at least 3 months after the last dose of study drug. * Note: The following are considered effective contraceptives: oral contraceptive pill; condom plus spermicide; diaphragm plus spermicide; patient or partner surgically sterile; patient or partner more than 12 months postmenopausal; or injectable or implantable agent/device. Male patients should refrain from sperm donation and female patients should refrain from breastfeeding throughout this period * RANDOMIZATION (STEP 2): Completion of the lead-in treatment of 6 cycles of ovarian function suppression (OFS) with use of goserelin, with estradiol E2 value of ˂ 20 pg/mL and no back to back E2 levels \> 10 pg/mL after cycle 6 blood draw Exclusion Criteria: * REGISTRATION (STEP 1): Any of the following prior therapies: Chemotherapy =\< 6 months prior to registration. NOTE: concurrent receipt of human epidermal growth factor receptor 2 (HER2) directed antibodies or antibody-drug conjugates, endocrine therapy, or cyclin-dependent kinase (CDK) 4/6 inhibitor is permitted * REGISTRATION (STEP 1): Receiving any estrogen or progestin containing medications, including topical estrogens * REGISTRATION (STEP 1): Planning to temporarily or permanently discontinue medical OFS in the next 12 months