The protection prevents the government from denying you the ability to make the threat. It does nothing to prevent them from pursuing a criminal prosecution after you have made it, or to prevent your victim from litigating a tort claim against you for intentional infliction of emotional distress.
I suppose the 2nd amendment prevents the government from denying me the ability to use a gun to shoot someone too... but saying that isn't constructive to the discussion of how to do we stop people shooting each other with guns.
Yes, I have the right to threaten someone because it's free speech, but I should still be punished for doing it.
(i.e. I have the right to have a gun and use it to shoot someone, but I should still be punished for doing it)
Basically, if an action could be considered a check upon government power, the government is not allowed to prevent people from doing it in the name of fighting crime.
You are allowed to have the gun that you might use for murder, because you might also use it to rebel against a corrupt government or in self-defense. You are allowed to use common channels of communication to make unlawful threats, because you could also use them to spread ideas unpopular to the ruling class. Only in the case where there is no possible constructive use or government-limiting use would the government be justified in engaging prior restraint.
As you might imagine, such a situation is exceedingly rare. A device designed to spread anthrax spores in an aerosol, for instance, has no peaceful purpose, and I would not consider it a legitimate check upon government power, as it would indiscriminately kill civilians as well as government agents. So you could be prosecuted just for having it, because it is presumed that it could not be used for any purpose that is not inherently criminal.
If there is any possibility that you might do something with innocent intent, the state has to wait until you do that thing with ill intent before punishing you for it. Now, there are a lot of laws and precedents that defy that principle, on a continuous spectrum from things I consider barely reasonable to things I consider to be completely unjust BS, so don't mistake any of this for real legal advice. What should be and what actually happens are often very different things.
