There are projects that make running independent atproto networks "easy": https://github.com/verdverm/testnet
I no longer recommend ATProto, in part because the public by default was a terrible choice. People prefer privacy, not anyone in the world able to read all of their activity. Bolting permissioned buckets on after the fact is not the way, it needs to be core to the protocol design.
I just started looking at the At Protocol for another side project - do you think the protocol will eventually support such privacy settings by default, or is heading in that direction?
It's baked in as deep as it can go.
Use a different protocol.
Read the "perm'd data diaries": https://dholms.leaflet.pub/3mhj6bcqats2o - It sounds like they have already picked this rough sketch as the basis.
My take is that (1) public vs private will be an app level choice, and user if the app passes that choice through and (2) this sketch is insufficient for many applications, being on the simpler side of the design spectrum.