@takeshitakenji If you're logged in with multiple clients, they can all see your own side of the conversation (otherwise only the client sending the message would see it). Also incoming messages get copied to clients that request carbon copies.
I.e. if you're switching clients often, maybe going from the computer to the phone and back again, then the full conversation will be available on both devices.
(this is not MAM, message archive management, which is for retrieving conversations if you've been offline. carbons requires you to be online)