It's completely local unless you specifically opt into cloud options. There is Obsidian Sync but that's completely optional, and your files are still on the computer. I know some people make their vaults Google Drive folders, which, again, is something you have to deliberately do.

100% non-cloud. There are sync options but they are completely optional. No log-in required unless you use the cloud features.

Obsidian in my experience is zim but better (at the cost of being closed source)