I've been testing 22H2 via Insiders Beta channel for a couple of months now; the Live Captions feature is great and has been almost life-changing for me. Really glad it's rolling out to wider availability now.
That said, Win11 live captions work on-device and are not server-bound, and I've confirmed this by playing audio files and having captions still work while entirely off the internet.
For that matter, while I'm not an Android user - Google has moved most of their captions processing on-device and that is the same approach Apple is taking with theirs.