It is open source and everything is stored locally but not in markdown files. You can self-host the desktop app to share notes.
Also, is real time collaboration a game changer or a gimmick?
I used to think the former, but was questioning it recently.
What would be lost with syncthing + obsidian for your team or maybe something that does autocommit like fossil for probable conflicts?
The real time is not as real time as, say, Google Docs. There's a lot of lag.
But yes, realtime collaboration is a critical feature for documents that can be used during meetings (and IMO ~no meetings should be held without documents).
The other critical feature is comments, which I assume Obsidian does not have support for (they would be very nontrivial to represent in markdown).
https://gist.github.com/rattrayalex/2e4f934045aabc4caeeab249...
I really wish Obsidian, or something like it, worked well for teams. Maybe it's on their roadmap, I'm not sure.