In lieu of a official official answer - I'm going to try to go for a credible one.
I think the canonical post on "suicide posts" is a good place to start off my answer
This isn't a support group; y'all probably aren't trained to deal with the outpouring of grief and despair of someone you've never met and may have absolutely nothing in common with.
I think the same goes as far as abuse goes. We arn't prepared (as users and moderators) to deal with the grief, despair and pain of total strangers.. The most important place to start is harm reduction. I do realise there's sites (these days) where this sort of thing is on topic - but as a user I'd ask myself "is this helping the other user?" Then flag anyway, so a moderator can take a look.
While as a moderator on a tech site, I'm unlikely to come across a credible, on topic case of this - I'd likely try to decide how urgent this is. While there's no canonical set of resources I'm aware of for abuse cases - I'd probably throw a lock on the question, letting the user know we're checking out how best to help them, and pass it up to the CM team. If its something urgent, chances are we'll grab a CM on chat where possible - bypassing the 'usual' channels.
Presumably they have an idea of what to do, or at least resources and official weight behind their actions.
Yeah, all this sounds a bit vague. In situations like this, every case is different.Unlike facebook or twitter, SE has never been a 'social network' and sites where such things would even be in scope are relatively new. As a moderator on a tech site (which SE was originally built around), I think I've seen less than 2-3 'fake' suicide or abuse claims, and no real ones.
Likewise - looking at best practices from those spaces, facebook basically ... just throws you a webpage and a pamphlet. Having reported abusive content in the past, its felt a little more impersonal. No one really has the sort of super warm, fuzzy help without being specialised in that sort of thing I suspect.
And practically - there's little that can be do to directly help someone in that sort of trouble. SE's mostly US based, and the right place to go for help would be dependant on where they are. What SE can do directly is probably limited, so while you're talking to a real SE employee, its not like they can go "You should talk to this person". They're lovely folks but chances are they'll need to figure out what best to do based on the situation and the circumstances of the user and staff member.