Docetaxel, Carboplatin, Trastuzumab, and Pertuzumab With or Without Estrogen Deprivation in Treating Patients With Hormone Receptor-Positive, HER2-Positive Operable or Locally Advanced Breast Cancer

Inclusion Criteria: * Patients should have a life expectancy of at least 10 years, excluding their diagnosis of breast cancer; (comorbid conditions should be taken into consideration, but not the diagnosis of breast cancer) * Women of reproductive potential must agree to use an effective non-hormonal method of contraception during study therapy (chemotherapy, trastuzumab, pertuzumab, and estrogen deprivation therapy) and for at least 7 months after the last dose of study therapy * Submission of tumor samples is required for all patients; the local pathology department policy regarding release of tumor samples must be considered in the screening process; patients whose tumor samples are located in a pathology department that by policy will not submit any samples for research purposes should not be approached for participation in the B-52 trial * The patient must have signed and dated an Institutional Review Board (IRB)-approved consent form that conforms to federal and institutional guidelines * The patient must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Clinical staging for the primary tumor can be cT1c (must be 2.0 cm) or T2-T4 if clinically node negative; if the regional lymph nodes are cN1 and cytologically or histologically positive or if cN2-N3 with or without a biopsy, the primary breast tumor can be cT0-T4 * The diagnosis of invasive adenocarcinoma of the breast must have been made by core needle biopsy * Ipsilateral axillary lymph nodes must be evaluated by imaging (mammogram, ultrasound, and/or magnetic resonance imaging \[MRI\]) within 6 weeks prior to randomization; if suspicious or abnormal, fine needle aspirate (FNA) or core biopsy is recommended, also within 6 weeks prior to randomization; findings of these evaluations will be used to determine the nodal status prior to randomization: * Nodal status - negative * Imaging of the axilla is negative * Imaging is suspicious or abnormal but the FNA or core biopsy of the questionable node(s) on imaging is negative * Nodal status - positive * FNA or core biopsy of the node(s) is cytologically or histologically suspicious or positive * Imaging is suspicious or abnormal but FNA or core biopsy was not performed * Patients may be premenopausal or postmenopausal at the time of randomization; for study purposes, postmenopausal is defined as: * Age 56 or older with no spontaneous menses for at least 12 months prior to study entry; or * Age 55 or younger with no spontaneous menses for at least 12 months prior to study entry (e.g., spontaneous or secondary to hysterectomy) and with a documented estradiol level in the postmenopausal range according to local institutional/laboratory standard; or * Documented bilateral oophorectomy * The tumor must have been determined to be HER2-postive as follows: * Immunohistochemistry (IHC) 3+ or * In situ hybridization (ISH)-positive (defined by ratio of HER2 to circulating endothelial progenitors \[CEP\]17 \>= 2.0 or HER2 gene copy number \>= 6 per nucleus) * The tumor must have been determined to be estrogen receptor (ER) and/or progesterone (PgR) positive assessed by current American Society of Clinical Oncology (ASCO)/College of American Pathologist (CAP) guideline recommendations for hormone receptor testing; patients with \>= 1% ER or PgR staining by IHC are considered positive * Absolute neutrophil count (ANC) must be \>= 1200/mm\^3 * Platelet count must be \>= 100,000/mm\^3 * Hemoglobin must be \>= 10 g/dL * Total bilirubin must be =\< upper limit of normal (ULN) for the lab unless the patient has a bilirubin elevation \> ULN to 1.5 x ULN due to Gilbert's disease or similar syndrome involving slow conjugation of bilirubin * Alkaline phosphatase must be =\< 2.5 x ULN for the lab * Aspartate aminotransferase (AST) must be =\< 1.5 x ULN for the lab * Alkaline phosphatase and AST may not both be \> the ULN; for example, if the alkaline phosphatase is \> the ULN but =\< 2.5 x ULN, the AST must be =\< the ULN; if the AST is \> the ULN but =\< 1.5 x ULN, the alkaline phosphatase must be =\< ULN; Note: If alanine aminotransferase (ALT) is performed instead of AST (per institution's standard practice), the ALT value must be =\< 1.5 x ULN; if both were performed, the AST must be =\< 1.5 x ULN * Patients with AST or alkaline phosphatase \> ULN are eligible for inclusion in the study if liver imaging (computed tomography \[CT\], MRI, positron emission tomography \[PET\]-CT, or PET scan) performed within 6 weeks prior to randomization does not demonstrate metastatic disease and the requirements are met * Patients with alkaline phosphatase that is \> ULN but =\< 2.5 x ULN or unexplained bone pain are eligible for inclusion in the study if a bone scan, PET-CT scan, or PET scan performed within 6 weeks prior to randomization does not demonstrate metastatic disease * Within 6 weeks prior to randomization, the most recent serum creatinine must be =\< ULN or measured or calculated creatinine clearance must be \> 60 mL/min * Left ventricular ejection fraction (LVEF) assessment must be performed within 90 days prior to randomization; (LVEF assessment performed by 2-dimensional \[2-D\] echocardiogram is preferred; however, multi gated acquisition scan \[MUGA\] scan may be substituted based on institutional preferences); the LVEF must be \>= 50% regardless of the cardiac imaging facility's lower limit of normal; note: since the pre-entry LVEF serves as the baseline for comparing subsequent LVEF assessments, it is critical that this baseline study be an accurate assessment; if the baseline LVEF is \> 65%, the investigator is encouraged to have the accuracy of the initial LVEF result confirmed and repeat the test if the accuracy is uncertain Exclusion Criteria: * FNA alone to diagnose the breast cancer * Excisional biopsy or lumpectomy performed prior to randomization * Surgical axillary staging procedure prior to randomization; pre-neoadjuvant therapy sentinel node biopsy is not permitted * Definitive clinical or radiologic evidence of metastatic disease; (chest imaging \[mandatory for all patients\] and other imaging \[if required\] must have been performed within 90 days prior to randomization) * Synchronous bilateral invasive breast cancer * Synchronous or previous contralateral invasive breast cancer; (patients with synchronous and/or previous contralateral ductal carcinoma in situ \[DCIS\] or lobular carcinoma in situ \[LCIS\] are eligible) * Any previous history of ipsilateral invasive breast cancer or ipsilateral DCIS; (patients with synchronous or previous ipsilateral LCIS are eligible) * Treatment including radiation therapy (RT), chemotherapy, targeted therapy, or endocrine therapy for the currently diagnosed breast cancer prior to randomization * Previous endocrine therapy such as raloxifene or tamoxifen (or other selective estrogen receptor modulator \[SERM\]) or an aromatase inhibitor for any malignancy * Previous therapy with anthracycline, taxanes, carboplatin, trastuzumab, or other HER2 targeted therapies for any malignancy * Any sex hormonal therapy, e.g., birth control pills, ovarian hormone replacement therapy, etc. (these patients are eligible if this therapy is discontinued prior to randomization) * History of non-breast malignancies (except for in situ cancers treated only by local excision and basal cell and squamous cell carcinomas of the skin) within 5 years prior to randomization * Cardiac disease (history of and/or active disease) that would preclude the use of the drugs included in the treatment regimens; this includes but is not confined to: * Active cardiac disease: * Angina pectoris that requires the use of anti-anginal medication; * Ventricular arrhythmias except for benign premature ventricular contractions; * Supraventricular and nodal arrhythmias requiring a pacemaker or not controlled with medication; * Conduction abnormality requiring a pacemaker; * Valvular disease with documented compromise in cardiac function; and * Symptomatic pericarditis * History of cardiac disease: * Myocardial infarction documented by elevated cardiac enzymes or persistent regional wall abnormalities on assessment of left ventricular (LV) function; * History of documented congestive heart failure (CHF); and * Documented cardiomyopathy * Uncontrolled hypertension defined as sustained systolic blood pressure (BP) \> 150 mmHg or diastolic BP \> 90 mmHg; (patients with initial BP elevations are eligible if initiation or adjustment of BP medication lowers pressure to meet entry criteria) * Active hepatitis B or hepatitis C with abnormal liver function tests * Intrinsic lung disease resulting in dyspnea * Poorly controlled diabetes mellitus * Active infection or chronic infection requiring chronic suppressive antibiotics * Patients known to be human immunodeficiency virus (HIV) positive with a baseline cluster of differentiation (CD)4 count of \< 250 cells/mm\^3 or have a history of acquired immune deficiency syndrome (AIDS) indicator conditions; patients taking anti-retroviral therapy that may have a potential overlapping toxicity with the study therapy are not eligible * Nervous system disorder (paresthesia, peripheral motor neuropathy, or peripheral sensory neuropathy) \>= grade 2, per the Common Terminology Criteria for Adverse Events version 4.0 (CTCAE v4.0) * Malabsorption syndrome, ulcerative colitis, resection of the stomach or small bowel, or other disease significantly affecting gastrointestinal function * Other non-malignant systemic disease that would preclude treatment with any of the treatment regimens or would prevent required follow-up * Conditions that would prohibit administration of corticosteroids * Chronic daily treatment with corticosteroids with a dose of \>= 10 mg/day methylprednisolone equivalent (excluding inhaled steroids) * Known hypersensitivity to any of the study drugs or any of the ingredients or excipients of these drugs (e.g., polysorbate 80), including sensitivity to benzyl alcohol * Pregnancy or lactation at the time of study entry; (note: pregnancy testing must be performed within 2 weeks prior to randomization according to institutional standards for women of childbearing potential) * Psychiatric or addictive disorders or other conditions that, in the opinion of the investigator, would preclude the patient from meeting the study requirements * Use of any investigational product within 30 days prior to randomization