For it to be a criminal offense, the threat must be 'specific' and 'immediate'.
They must be able to carry out the threat, and must direct it towards your person in a specific manner. They must say something like, "I got a gun, and I'm coming to your house right now that's on 1222 Mulberry lane!" (or you know that they know where you live).
If they say something like, "I really want to kill you right now!" or "I hope you die in a fire!"
Those are not 'immediate' threats; they are not saying they are going to kill you right now, nor did they specify some kind of time frame.
Obviously the law is flexible, so if you feel that a valid threat has been made and that your life is indeed in danger, then going to the authorities is the best thing you can do.