Sealed on your computer, opened on your phone
Your computer and phone don't have to share a network. A laptop on office Wi-Fi reaches a phone on mobile data through Clair's relay, and the relay only ever handles sealed messages. The agent seals each prompt before it leaves your computer, and only your phone holds the key to open it.
To deliver messages and hold off abuse, the relay does see when each pairing is active, how much it sends, the IP addresses your devices connect from, and the push token that wakes your phone. From your phone, prompts reach the watch over Wear OS's own phone-to-watch connection, which runs through Google's servers when the two aren't within Bluetooth range.
Revoke a computer from the phone, or run clair unpair on the computer. Either one ends the pairing, and anything still waiting on the relay is deleted with it. A pairing also ends on its own if one side stays silent for 30 days.