Sometimes, a moderation decision does not go down well. This can lead the occasional user to take to Twitter, Facebook, even email to hurl abuse. These attacks can be incredibly personal. What course of action should be taken if this situation arises?
First piece of advice when contacted in such a way - ignore it. Don't respond positively ("I'm sorry you feel that way") or negatively ("get lost, idiot"). Don't feed the trolls. And also you don't want to confirm to them that they have found your details correctly.
If it persists for more than one occurrence then I would say you would treat such behaviour in the same manner as you would were it not moderator related. Namely - use the 'report abuse' option that is provided on the site the abuse came in on.
It is likely (though not guaranteed) that abusive behaviour is against the terms and conditions of the various sites you get abuse coming into you on.
Also an idea (something I myself did) is to remove links to your external personal social media sites from your personal profile. Anything that makes it harder for users to get in contact with you externally requires more effort on their part, therefore putting off all but the most vehement users.
Finally, from a more personal perspective; just try to laugh it off. The world is a large place, and just because someone on Twitter wants you to 'die in a fire' that doesn't mean they're either capable or willing to actually carry through with their threats. The chances are it's just a brief spell of rage on their part and you won't be contacted again by them, so just treat the abuse as a one-off, ignore it and walk away.
While JonW has some great suggestions, and I agree with them, I believe there is one missing element: Documentation!
Document everything. Record the fact that the user approached you using inappropriate channels. Share this knowledge with other moderators on your site. Place a notice on the user's account. Whatever it takes to ensure that you have an 'audit trail' that you can reference if you need to 'go public' at some point in the future. It also allows other moderators to handle things differently if they get contacted through personal channels, as it indicates a pattern, rather than a one-off.