Not too certain about this, but when you assign damage in Two Headed Giant you choose which player takes the damage. In this case, that player would get the poison counter, so in the future you would have to choose to assign damage to that same player and once that player has 10 counters, their team loses.
(Someone correct me if I'm wrong)
606.7d. As the combat damage step begins, the active team announces how each attacking creature will assign its combat damage. If an attacking creature would assign combat damage to the defending team, the active team chooses only one of the defending players for that creature to assign its combat damage to. Then the defending team announces how each blocking creature will assign its combat damage.
Well, that's a problem because the the players take damage as a team.
Either way, once one of the players (or both) get 10 poison counters, the team loses the game. (they should probably write this in the rules somewhere)