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.
Anytype as an alternative to Notion or Obsidian
Stumbled on this program called Anytype a while ago, a note-taking application similar to Notion. It’s surprisingly well polished and works for me....