Can’t say for sure. Everything is considered an object in anytype. As I understand it every line of text, every bullet point could be its own file. Instead of one markdown file for each page like obsidian does. How this will affect the syncing I can’t say.

Right now you can’t turn off the built-in sync with their servers (except by blocking traffic with a firewall). So there isn’t really anything to gain in hosting the files yourself.

They plan to implement 3rd party sync though and there seems to be a docker image to self host, that would likely be a better option. https://doc.anytype.io/anytype-docs/data-and-security/data-storage-and-deletion/self-hosting