I haven't had any issues with sync, I often have to work in areas without wifi/data and as amazed at the LAN aspect of hotspotting with this app, The backup feature is great for peace of mind aswell.
One thing to note is that it seems to sync (encrypted) to their server, with no option to turn it off [yet]
From my privacy trials on ollama - any model downloaded does not know the date or time and cannot access the internet.
If you are still sceptical you could download something like alpaca on flathub and once youve acquired a model, remove internet access etc through flatseal.
theres a cool preset called "futuristic" on the linux version (cool retro term) -with a bit of tweaking you can make it look like a terminal from the alien franchise
https://fcast.org/