I suppose I'd try to console them.. but with my other hand, I'm dialing 9-1-1 and asking for instructions on what the hell I should about the situation. At that point, I do whatever they tell me to do, and I stop caring; whatever happens is on the person who called me, and the police to a lesser extent; I'm simply a middle-man.
Although, this has actually happened to me before, albeit not with a stranger. A kid called me who I never got along with very well; he started thanking me for always being nice to him (I wasn't a bully or anything - we simply did not get along very well) and for always being his friend. This immediately set off alarm bells since he knew damn well that we were not friends, and he was also speaking in the past tense.. you know what I mean, I hope. I immediately told the teacher who actually knew his parents' number. Blah blah, he'd apparently been calling a lot of people, and I wasn't the only one who thought something was up.. blah blah.. parents found him, he's still alive. Fin.