[aoeusnth1]

How is the extreme version (no state) different from Ancaps who want private police, fire, roads & private ledgers of property?

Encouraging individuals or groups to have their own notions of who owns what seems like a surefire way for warlords & petty tyrants to take over local economies.

[int_19h]

It's different in the expectations from such a society by adherents. Ancaps believe that private property can exist in the absence of a state protecting it at scale. Anarcho-socialists think that private property is a social construct that requires the state to prop it up, and that would disappear if you removed the state.

There is a book by Ursula Le Guin, "The Dispossessed", that describes an utopian (to some extent; the book subtitle is "an ambiguous utopia" for a reason) anarcho-socialist society. Anarcho-syndicalist, to be precise. It's fiction, of course - but if you wanted to get the gist of how people who believe in this sort of thing imagine such a society would work, it's a great introductory crash course; kinda like "The Moon is a Harsh Mistress" is for ancap.