What happens with Poison Counters in Two Headed Giant?
In Two-Headed Giant you need 15 or more poison counters. See Comprehensive Rules.

704.5t In a Two-Headed Giant game, if a team has fifteen or more poison counters, that team loses the game. See rule 810, “Two-Headed
Giant Variant.”

Also

810.5. With the exception of life total and poison counters, a team’s resources (cards in hand, mana, and so on) are not shared in the
Two-Headed Giant variant. Teammates may review each other’s hands and
discuss strategies at any time. Teammates can’t manipulate each
other’s cards or permanents.

You share those poision counters as a team. So if your teammate is
being hit for 5 poison counters, your team has 5 poison counters.

810.8a Players win and lose the game only as a team, not as individuals. If either player on a team loses the game, the team loses
the game. If either player on a team wins the game, the entire team
wins the game. If an effect says that a player can’t win the game,
that player’s team can’t win the game. If an effect says that a player
can’t lose the game, that player’s team can’t lose the game.

You win as a team, you lose as a team. There is no 'I' in team.
Further clarification regarding poison counters in Two-Headed Giant.

810.10. Effects that cause players to get poison counters happen to each player individually. The poison counters are shared by the team.
810.10a If an effect needs to know how many poison counters an individual player has, that effect uses the number of poison counters
that player’s team has. If an effect needs to know how many poison
counters a player’s opponents have, that effect uses the number of
poison counters opposing teams have.
810.10b If an effect says that a player loses poison counters, that player’s team loses that many poison counters.
810.10c If an effect says that a player can’t get poison counters, no player on that player’s team can get poison counters.
810.10d If a rule or effect needs to know what kinds of counters an individual player has, that effect uses the kinds of counters that
player has and the kinds of counters that player’s team has. A player
is “poisoned” if his or her team has one or more poison counters.