The ones with mental illness should be identified.
Check this out:

The Obama administration says it is doing away with “unnecessary legal barriers” that discourage states from sending the names of some mentally ill people to the national background check system for gun-buyers.

Late on Friday afternoon, the Department of Health and Human Services announced that it plans to modify HIPAA — the Health Insurance Portability and Accountability Act of 1996 — “to expressly permit certain HIPAA-covered entities (hospitals, doctors, clinics, etc.) to disclose to the National Instant Criminal Background Check System (NICS) the identities of individuals” with mental problems that should legally disqualify them from owning a gun.