Yes, the "setup" messages via the Mailbox Server will be over the internet to a third-party server.
All the contents of these messages are end-to-end encrypted so you reveal which two IP addresses are communicating, but not the contents of those communications. (If you don't want to reveal that, use the Tor options).
The "bulk transfer" connection should be over the LAN only if both devices are on the same network. In any case, all of these messages are also end-to-end encrypted as well.
All the contents of these messages are end-to-end encrypted so you reveal which two IP addresses are communicating, but not the contents of those communications. (If you don't want to reveal that, use the Tor options).
The "bulk transfer" connection should be over the LAN only if both devices are on the same network. In any case, all of these messages are also end-to-end encrypted as well.