Originally Posted by Florya

Obviously you would check their medical records. That's what happens in the UK.
Of course you can't guarantee that a person won't develop mental problems in the future, and there is no way of knowing that the application for a gun license isn't the first sign of pending mental illness. That's why the application needs to be made public, so that people that know the applicant can be made aware of his/her intentions and object if they feel that allowing them to have a gun would be inappropriate.