It varies per department. There is no standard hiring process or even set of customs or rules for police departments across the country. Some states have various guidelines and laws regarding police management, and many counties have their own guidelines too, but for the most part there is no standard.
Is that a problem? Yeah, definitely. But it's also not an issue that a psych exam would solve, really. Cops are not evil, or psychotic. They're just arrogant, heavy-handed and often don't have good situational judgment skills - things that are gained through training and experience. The attitude of police in the United States is generally what causes issues, not their psychological status or whatever else.