He means PvP chat.
I think it's very possible that this could happen, but like teffy said, the added data could take more time to load on the other person's screen, which could ultimately lead to many more desyncs. Good idea, though.
Unless we are smart about it and only use avatars based on art already in the game.
I assume when I get the info from my opponent's turn it is all code which tells my client what to display to me. I am not actually getting the information for the whole card, just the info telling me what card and then my client fills in the rest.
In the same way, if we had set avatars based on existing art, it may be possible to have an avatar where very little information is sent.
If my avatar is mitosis, and I send a message - my opponent's client would simply get the (art) code for mitosis and display it based on what it already has.
Obviously custom avatars would be an annoying pain in the butt, but I think we could reasonably get avatars based on already existing in game art without the data problem.
