Leuprolide Acetate or Goserelin Acetate Compared With Observation in Treating Patients With High-Risk Prostate Cancer Who Have Undergone Radical Prostatectomy

Inclusion Criteria: * PRE-REGISTRATION: * Informed consent explained and signed prior to any study related procedures * Patients with any one of the following "high risk" criteria: * Clinical or pathological Gleason score 8-10 * Prostate-specific antigen (PSA) \> 20 ng/ml at initial presentation prior to radical prostatectomy * Willingness to provide mandatory tissue for research purposes * Willingness to provide mandatory blood for research purposes * Has no history of androgen deprivation therapy within the past 6 months or has been treated neoadjuvantly up to 6 months prior to radical prostatectomy with the following agents; luteinizing hormone-releasing hormone (LHRH) agonists, anti-androgens, 5 alpha-reductase inhibitors, and peripheral anti-androgens * REGISTRATION: * Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1 or 2; or Karnofsky performance of \> 60% * Patients with any one of the following "high risk" criteria: * Gleason, prostate specific antigen, seminal vesicle and margin status (GPSM) score \>= 10 \[GS + 1\*(PSA 4-10)+2\*(PSA 10.1-20)+3\*(PSA \> 20)+2\*(seminal vesicular or nodal involvement) +2\*(margin)\](determined post radical prostatectomy) * Post prostatectomy seminal vesicle invasion (pT3b) or pT4 * Two or less microscopic lymph nodal metastasis determined at the time of prostatectomy OR * Gleason 4+3 at the time of prostatectomy with margin positivity * Aspartate aminotransferase (AST)/serum glutamic oxaloacetic transaminase (SGOT) =\< 2 x institutional upper limit of normal (ULN) * Total bilirubin =\< 2 x institutional ULN * For patients identified as high-risk on the basis of pathological criteria after undergoing radical prostatectomy: interval time for study enrollment after radical prostatectomy will be =\< 28 days of the prostatectomy * For patients identified as high-risk prior to undergoing radical prostatectomy: patients presenting with a high Gleason score (8-10) and/or a PSA \> 20 ng/ml are deemed eligible for study participation and study registration as long as the eligibility criteria is reconfirmed post radical prostatectomy; these patient groups may choose to register prior to or after prostatectomy * Study randomization must occur =\< 28 days of radical prostatectomy; all patients consented on the trial, whether consented in the pre-prostatectomy or post-prostatectomy period, will be randomized to study treatments =\< 28 days of prostatectomy * Ability to complete questionnaire(s) by themselves or with assistance Exclusion Criteria: * PRE-REGISTRATION * Transitional cell, small cell, or squamous cell carcinoma of the prostate; NOTE: patients consented for participation prior to prostatectomy, if detected to have above listed histo-pathologies after prostatectomy will be deemed ineligible and not proceed to study randomization * History of primary prostate cancer treatment * Evidence of clinical nodal disease (N1) or grossly evident metastasis at the time of enrollment * History of bilateral orchiectomy; unilateral orchiectomy with normal range serum testosterone levels will be allowed for enrollment * Evidence of metastasis on radiographic metastatic workup within a preceding period of 4 months from the time of study entry, including whole body radionuclide bone scan, computed tomography (CT) and/or magnetic resonance (MR) scan of the pelvis and abdomen; otherwise will perform at the time of the baseline tests and result must be normal to continue on study; results of ProstaScint or other radionuclide scans, excluding radionuclide bone scans, will NOT be used to establish metastatic disease if all other studies are negative * Receiving other experimental drugs =\< 4 weeks prior to consenting * Uncontrolled infection * History of other cancer, excluding squamous cell and basal cell skin cancers, within the preceding 2 years * Documented history of human immunodeficiency virus (HIV) positivity or other acquired immunodeficiency disorder, congenital immunodeficiency disorder, or history of organ transplantation * Unable to follow up every three months for the first year to Mayo Clinic, Rochester for receiving LHRH analogues or study monitoring * REGISTRATION: * Uncontrolled infection * Unable to follow up every three months for the first year to Mayo Clinic, Rochester for receiving LHRH analogues or study monitoring