You can read Obsidian's privacy policy. Basically, everything remains on your device unless you pay to use the Obsidian Sync. I switched from Standard Notes to Obsidian last year and I haven't looked back ever since. You can use Syncthing to synchronize your Obsidian Vault across multiple devices. All you need to do is add the Vault directory to Syncthing, that means you need to first make a dedicated folder in your filesystem for the Obsidian Vault which you will be required to do anyway while setting up Obsidian.