I visited Freetown Christiania in Copenhagen when I traveled to Denmark. I don't have a lot of information, but I walked by someone giving a tour, and from the little I overheard, it sounded like that's how it worked there. Everyone rotated handling the various responsibilities to take care of what needed to be done. But that's a small community of like-minded people, so it doesn't really have to scale like it would for it to work on a large city, state, or country level.
I've often wondered if the solution to the "who will clean the toilets?!" problem is: everyone, but only when it's their month to do so.
A pipe-dream, of course, and likely only doable under a military dictatorship.