I feel private (intra-team) chatting will be the death-knell of team play.
1) A single player controlling both sides has an innate advantage over two opponents controlling their own respective sides (the single player has perfect/instantaneous/effortless communication with "partner").
2) Eventually (if it hasn't already happened), an already-strong 4-PC player is going to find a willing "puppet" partner.
3) The team will be controlled completely by the puppet-master, who will call out moves to the puppet in private chat.
4) Both puppet master and puppet benefit: the "team" will be virtually unbeatable (with the exception of other "puppet" teams who might have a better puppet master); both will quickly rise to the top of the leader board.
I should note - this scenario is the "end-game", where I think things will wind up. However, symptoms of it already run rampant. My last 4PC team game, I was a 1200-rated player matched with 1450-rated player. This partner called out every move from move 1. Each move made good sense, and partner *was* high-rated, so I followed his suggestions. We checkmated in < 10 moves.
This was the least-fun and most demoralizing win I've ever had in chess... in fact, so bad a taste did it leave that I haven't played a game since. "Just ignore the suggestions", you might say. That's like working on a Sudoku puzzle while a Sudoku expert is helpfully telling you what values go in each box.
I think team-4PC should be more like the card-game "bridge", wherein ANY game-related/strategic communication between teammates is considered cheating. The only communication in bridge is communication via how one plays. Far from detracting from play, this forced-radio-silence is actually one of the best and most appealing features of bridge; great teams are those who are able to infer what their partner is up to and act in a synergistic fashion.
I should note, even disabling "private chat" mode doesn't totally solve this problem. The puppeteering team still has the exact same advantage if puppet-master communicates puppet's moves on "/all" chat (that is, the puppet master can still call out moves... yes, everyone will see that, but if there aren't rules against it, what difference does it make?). To that end, any communication which in any way hints at what a teammate should do should be considered cheating (and should be flag-able by opponents such that routine cheaters can be penalized/weeded out). Yes, processing "cheat accusations" adds a layer of complexity to game-design; but there is precedence for this. Online bridge exists; I'd recommend studying and emulating the controls used to discourage casual cheating in that realm.