You can't screen the people who own illegal firearms, they own them that way for a reason. As far as the people who already own guns, I'm not sure how that would pan over. In light of recent events I think it'd be beneficial to take a psychological exam before getting a firearm permit. I also propose that your driver's license test should have to be retaken every 5 years. It's plain and simple, the only ones who would complain about such measures would be the ones not qualified in the first place.
There is a right to bear arms under the Constitution but driving is only a privilege. A right and a privilege are quite different, especially when it comes to regulating them.
Thus, you could retest drivers every 5 years or give psychological exams to fox's example of Mercedes owning wives of cheating husbands (still my favorite).
Establishing a psychological test or even a basic level skills test to purchase a weapon has never been upheld. Remember the people who are prohibited from purchasing weapons now have given up that right by being found incompetent or committing a felony.