With signal each registered device will receive a copy of the chat while that device is registered. If a new device is enabled afaik it doesn't get sent any messages from the past, although technically the other devices could transmit previous messages.
As far as I can tell, the compression is separate for the headers, does not use deflate, and (most importantly) is over the entire session, not just one request.
This would benefit from the fact that in one request headers are not repeated, but over multiple requests they certainly are.