The United States is that exception, because our constitution explicitly recognizes, not grants, our right to arm ourselves independently of the State (or States really), only with some regulation permitted.
The actual definition of State can easily be found in the dictionary: "a nation or territory considered as an organized political community under one government."
Also it's obviously implied that the right to bear arms includes the right to use them under at least some circumstances. It would just be as ludicrous as saying freedom of the press only covers your right to own a press in your house or business and not your right to print things with it.
And in doing so, they are acting as self-deputized agents of the state. They're temporarily "arming" themselves with the state's power to use violence in furtherance of its (legitimate) interests in enforcing its laws. They could not legally do that without the state's sanction.
EDIT: Some countries don't even limit the power of "citizen's arrest" to their own citizens. The UK's relevant statute specifically says "any person", emphasis added, may act thusly.