I'm not a fan of mental health checks for a multitude of reasons. The first is that it's ineffective. What you're really trying to prevent is something getting into the hands of a sociopath vs. someone with legitimate mental health problems. Which leads me in to my second reason, that people always make a push for more access to mental health treatment any time there's a national tragedy. This is nonsense. It further stigmatizes the mentally ill by continuing to draw associations with people who are simply sociopaths and not suffering from some other mental illness. There's no therapy or medication you're going to give a sociopath that's going to prevent, to any degree, any of their sociopathic actions and behaviors. If a guy has schizophrenia and believes that his neighbor is conspiring with the KGB to kill him, it could certainly pose a danger to that individual, and you could treat the psychosis, but unfortunately these aren't the situations we're seeing -- we're seeing sociopaths who like harming people -- that's not a "chemical imbalance." On another note, though many sociopaths do find their way into treatment, creating a mental health check won't catch the sociopaths -- it will catch all the other people who are much more likely to be victims of violence than perpetrators -- and we can further stigmatize them. All the proposed Muslim restrictions that people bring up wouldn't prevent terrorism, and would only continue to stigmatize regular Muslims -- the same is true with applying this to mental health (though this is currently a huge blind-spot for people who are normally against this kind of thing).
Regarding whether or not a physician could tell law enforcement regarding someone who may harm someone, there is some small variability state-to-state. If there is a specific threat or target, most states have a duty-to-warn law, while some others have a right [not duty] -to-warn. I used to practice in Texas, which was a right-to-warn state. I could inform law-enforcement of a threat without breaking HIPAA (another layer bureaucracy that makes everyone feel good about protection but more often gets in the way), but I couldn't inform the actual individual without violating HIPAA laws. I had a situation where there was a kid who was admitted to the hospital for nothing more than being a little sociopath (i.e. he didn't really have something we could treat) because he was threatening to kill a girl. I reported this to the police of the town, who didn't want to handle it because he wasn't from there, I reported it to the county sheriffs office, who also didn't want to take care of it, and eventually the police officer in his small town, who eventually told the girl's family. The bottom line was that everyone assumed, because he was hospitalized in a psychiatric facility, that he said these things "because he was crazy" and that "we wouldn't let him out until he's all better." Bottom line is we need to stop conflating the mentally ill with sociopaths.
