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.