In case anyone is interested, here are some quick tests I ran to check internet usage of the desktop app (Windows).
First I tested using the app only on my local KB (with a few MB). As expected, no data seems to have been sent out. There was around 0.1MB (probably less) sent/received at every restart of the app, but I guess this is just update checks (?).
PS: first process in the image is the latest Setup exe. Since I cleared the usage history before doing this, it has 0MB usage.
Then I used the app to connect to my online account and pasted some images (both in the app, and in the web). The internet usage reflected that accordingly:
Hopefully this is helpful for those concerned with privacy (though of course this is just a superficial test).