Remember that well: what gives them power is an unerring force to use deadly force at a whim. "Make pot brownies using hash oil? No knock warrant with potential life sentence"...
Sure, an individual ceasing to believe in the power of a state is a drop in the ocean, and ultimately will not end well for the individual in most cases.
If a mass of individuals, however, cease believing, then the state's power evaporates. Witness every revolution, ever. It's not a question of the people assuming power, rather just the people refusing to grant the state power, en masse.
Also you talk about the state's power evaporating, but a shitload of peasant rebellions have been put down by those in charge. Like, almost all of them ever. Even the USA would have been utterly demolished if the Brits hadn't said "you know what, these colonials don't actually have enough value to fight over this. And they'll come crawling back anyway once they figure out they can't trade because Britannia rules the waves."