Acolbifene Versus Low Dose Tamoxifen for the Prevention of Breast Cancer in Premenopausal Women at High Risk for Development of Breast Cancer

Inclusion Criteria: * Age \>= 35 years * Considered clinically premenopausal * Having regular menstrual cycles (between 21 and 35 days) unless a contraceptive device such as progestin containing intrauterine device (IUD) (e.g., Mirena IUD) is being used which suppresses menstrual periods, or premenopausal women who have undergone a hysterectomy, but ovaries are intact * Not considering pregnancy for at least 12 months * Women of child-bearing potential capacity must be willing to have used effective birth control precautions for 8 weeks prior to fine needle aspiration and be willing to continue for 8 weeks after study completion as tamoxifen may have teratogenic effects on the developing fetus. Reproductive and developmental toxicity studies have not been conducted with acolbifene. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must stop study drug and inform her study physician immediately. * For women not using oral contraceptive (progestin alone or estrogen plus a progestin), two of the following are recommended but woman must agree to at least one of the following methods: * IUD non-hormonal or hormone containing (usually a progestin) intrauterine device (IUD) or rings. Any of these should have been inserted at least 8 weeks prior to RPFNA. * Barrier method (such as condoms and diaphragms or cervical caps with or without a spermicide) * Partner has had a vasectomy. * For women using oral contraceptive (progestin alone or estrogen plus a progestin), woman must agree to at least a non- hormonal IUD or a barrier method (below) or her partner must have had a vasectomy: * Non-hormonal IUD * Barrier method (such as condoms and diaphragms or cervical caps with or without a spermicide) * Partner has had a vasectomy * Must have increased breast cancer risk as predicted by any one or more of the conditions listed below or increased model calculated risk as below: * Any one or more of the following conditions associated with increased risk (condition must be documented in electronic medical record or copy of relevant pathology or genetic testing reports submitted with the eligibility checklist) * A prior biopsy at any time in the past showing ductal carcinoma in situ (DCIS), lobular carcinoma in situ (LCIS), atypical hyperplasia. (If DCIS must have been treated by mastectomy or local excision +/- radiation with this treatment completed at least 3 months prior to screening with RPFNA) * High or moderate penetrance risk pathogenic or likely pathogenic germline gene mutation in ATM, BARD1, BRIP1, CDH1, CHEK2, MSH6, NBN, NF1, PTEN, PMS2, RAD51C, RAD51D, or TP53 * High polygenic risk score (Life-time risk of \>= 2x average or 25%) * Breast cancer in a first or second degree relative (female or male) with onset under age 50. First degree relative is defined as parent, sibling, or child. Second degree relative is defined as grandparent, uncle, aunt, nephew, niece, half-sibling, grandchild or first cousin * Two or more affected first or second-degree relatives from either the maternal or paternal lineage without regard to age * Bilateral breast cancer or breast and ovarian cancer in the same first or second degree relative without regard to age. * High mammographic density defined as either visual estimate of area of density (VAS) \> 50%, or Volpara (Trademark) \>= 15% dense volume (Volpara d) or Breast Imaging Reporting and Data System (BIRADS) assessment of extremely dense (BIRADs D) * Alternatively, instead of conditions listed above, an increased risk of breast cancer as calculated by International Breast Cancer Intervention Study Version 8 (IBIS 8), or Breast Cancer Surveillance Consortium (BCSC) 3 by one or more of the following criteria: * 10-year risk of breast cancer of \>= 3% * Increase in age specific 10-year relative risk by age group * Age 35-39 10-year relative risk of \>= 5X that for age group * Age 40-44 10-year risk of \>= 4X that for age group * Age 45 and up 10-year risk of \>= 2X * IBIS Version 8 Remaining lifetime risk of \>= 25% or \>= 2X that of population * A copy of the output of model calculations from IBIS 8 (https://ems-trials.org/riskevaluator/), or BCSC version 3.0 (https://tools.bcsc-scc.org/BC5yearRisk/calculator.htm) online tools, if used for qualifying risk assessment, or polygenetic risk score should be submitted with the eligibility checklist. Otherwise, these risk qualifying factors need to be documented in the medical record if that is considered the source document * Women must have at least 1 unaffected untreated breast for fine needle aspiration. Women may have had prior unilateral breast radiation or mastectomy for DCIS * Eastern Cooperative Oncology Group (ECOG) current performance status (PS) ≤ 2 as documented within 3 months prior to randomization or Karnofsky score \>= 60% * Total bilirubin =\< 1.5 x institutional upper limit of normal (measured within 180 days prior to randomization) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) =\< 1.5 x institutional upper limit of normal (measured within 180 days prior to randomization) * Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 1.5 x institutional upper limit of normal (measured within 180 days prior to randomization) * Creatinine =\< 2.0 mg/dL (measured within 180 days prior to randomization) * Willingness and ability to comply with scheduled visits, treatment plans, laboratory tests, and other study procedures * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Bilateral breast implants (danger of implant puncture with RPFNA) * Women who are pregnant * Currently breastfeeding (concern that tamoxifen or acolbifene may be in breast milk) or nursing within the past 12 months (concern about milk fistula with RPFNA) * Prior invasive breast cancer within the past 5 years * Other prior invasive cancer \> T1 stage (other than non-melanoma skin) within the past 5 years * Pathogenic or likely pathogenic germline mutation in BRCA1/2 or PALB2 (These latter individuals are likely to undergo yearly ovarian screening and enlarging cysts could raise concern about ovarian cancer and lead to unnecessary diagnostic procedures) * Type I or Type II diabetes mellitus requiring treatment with prescription medication * Prior deep vein thrombosis, pulmonary embolus, or stroke * History of chronic liver disease including NASH (nonalcoholic steatohepatitis) and chronic hepatitis C * History of chronic hepatitis B or hepatitis C (danger of exacerbation of liver damage from hepatitis or tamoxifen-induced non-alcoholic fatty liver disease or non-alcoholic steatohepatitis) * History of human immunodeficiency virus (HIV)-infection (danger of exacerbation of underlying clinically inapparent liver damage caused by HIV and/or hepatotoxicity can be induced by interaction of tamoxifen-induced CYP3A4 with direct anti-hepatitis C virus \[HCV\] agents) * Current use of prescription anticoagulants such as Coumadin (warfarin), direct-acting oral anticoagulants such as Xarelto (rivaroxaban) or Eliquis (apixaban), or heparin * Women who would not be able to or do not wish to discontinue daily use of aspirin (81 mg or higher) and aspirin containing products (81 mg or higher) at least 3 weeks prior to each RPFNA are not eligible. Women who would be able to stop daily use of aspirin and aspirin containing products at least 3 weeks prior to each RPFNA are eligible * NOTE: Women may resume daily use of aspirin and aspirin containing products 3 days after each RPFNA procedure * Starting or stopping oral contraceptives (OCs) or hormonal progestin IUDs within 8 weeks of baseline RPFNA * Current use or use within the prior 8 weeks of progesterone/progestin injections or progestin implants (due to concerns about high levels of progestin and lack of safety and efficacy data with low dose tamoxifen) * Current use of other investigational agents * Prior treatment with acolbifene for more than 2 months * Prior treatment with tamoxifen for more than 2 months * Current use of prescription immunosuppressive drugs * History of allergic reactions attributed to tamoxifen or acolbifene or compounds of similar chemical composition * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study