Hereabouts, suicide attempts are responded to by Fire/EMS, and the Sheriff. Both agencies have received training in how to deal with these situations . We generally have the crisis folks on the radio/phone supporting us - there are few of them, they don't staff to respond in 60 seconds, and they generally aren't on all the islands.
The Sheriff is there primarily for scene safety. If the suicidal person is armed, the Sheriff deals with that part of the situation before the Fire/EMS people get to work.
I've responded to lots and lots of these cases, and I haven't seen the Sheriff shoot anyone yet, even when the situation started out with the person waving around a firearm.