Out of the box, they don't work offline. But there's recently been a project showing it's possible to create a PWA with CRDT's and LiveSvelte:

https://github.com/tonydangblog/liveview-svelte-pwa

Liveview does not work at all while offline. If you just experience a short temporary drop it can reconnect you automatically.

It's the same as any other web page, it doesn't work when you receive or send data from the server.

Liveview isn't that special, the liveview paradigm works best for what would already be online actions in a normal page.