I'm quite different in-game actually. There is not alot of talk, more action than anything else. I'm mostly recognized by the trail of dead bodies left behind me...
You know that old saying...
If you cannot beat em, join em?
I have alot of followers =)
I don't know how bnet 2.0 will do it, but in this game heroes of newerth I play right now, they have a system where people can message you via those text box things you were mentioning.
They don't get in the way, because of how they are set up.. they don't just appear on the screen your playing on, they appear on a interface that you press F6 to get into..
It's a interface that appears above your game screen when you play it and you can chat with people in the lobby you were in before joining a game(clan channel etc), check stats of players easily and stuff and your friend list and those chat text boxes are also there..
Also there is 1 window, on the left side of the box, there is a list of names.. jsut click that name to open up the chat of that person in the same box.. so it's not cluttered.
And once you get bored of talking or need to get to your game, press F6 again and you go back into the game interface.. it will say you get more messages on the bottom right of your screen, but you don't have to look at them just then.
I can only hope blizz is doing something like that and not how you described it to be.