Obsidian is pretty good, it shouldn’t collect any data by default. But you can also check out Logseq, an open source Obsidian alternative.

As they are closed source no one can tell you their true privacy policy. It seems better than average from what I’ve read but you never know…

Personally I use logseq and sync the files via a Nextcloud instance. I can only recommend it, although I also recommend spending an hour to learn the tagging and linking logic and reading through their guide on what’s possible. I still only leverage a minor part of the potential myself.

One that is closer to onenote (I think, never used onenote) is Joplin.

Since you’re specifically looking to replace OneNote, you might want to take a look at BookStack. It has similar organizational concepts, and I think it’s FOSS.

I use logseq for work notes and Obsidian for personal. Obsidian is more markdown which I like for my loose notes. logseq, on the other hand, is more focused on productivity and it’s fully opensource. Obsidian is only free for personal use, however their notes being closer to standart markdown means that they could be openned with any text editor and be just as functional.

Syncing between computers is easy – it’s just a git repo. Dealing with mobile is tricier but I never needed it so can’t comment much.

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.