The notes are stored in a SQLite database (which is just a file) in your Firefox profile directory. No data is transferred to the web. If you enabled synchronization, then your notes are also stored encrypted at the synchronization server.
Go to your Firefox profile folder and copy the file floatnotes.sqlite wherever you want to. You can always copy the file back but every new notes will be lost.
Sometimes, we don't know exactly when, your note might look like this:
This is "just" a rendering issue. It should be fixed be reloading the notes or the page.