Psychologists typically get an academic masters or doctors degree in psychology. They only get a license if they want to treat patients.
But I'm not sure there is any formal "screening" for mental illness in either process. I would think, however, it would be pretty tough for a nutjob to make it through internship and residency without someone noticing.
Ponder this, though. James Holmes was in a highly selective neuropsych program when he shot up the Aurora movie theater.
Wasn’t the Fort Hood shooter a shrink, as well?