Peer Education and Emergency Response for Mental Health Act of 2023 or the PEER Mental Health Act of 2023 This bill establishes grants for school-based programs for training and education focused on childhood and adolescent mental health. Specifically, the Department of Health and Human Services (HHS) must award the grants to elementary or secondary schools or state or local educational agencies. Recipients must use the grants to provide training and educational resources about childhood and adolescent mental health conditions to (1) teachers and other school personnel, (2) students, and (3) parents and caregivers. HHS must allocate at least 25% of the grants to schools in rural areas.