In a word, no.
Players do not share control of permanents in Two-Head Giant.
In addition, with the exception of "attacking player" and "defending player," all of the referents in THG are the same as regular Magic. "You" is you. "Target player" and "target opponent" refer to a single player. "Your opponents" are two players, individually.
Here are the relevant comp rules:
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.
810.7b Any one-shot effect that refers to the “defending player” refers to one specific defending player, not to both of the
defending players. The controller of the effect chooses which one the
spell or ability refers to at the time the effect is applied. The same
is true for any one-shot effect that refers to the “attacking
player.”
Any characteristic-defining ability that refers to the " “defending player” refers to one specific defending player, not to
both of the defending players. The controller of the object with the
characteristic-defining ability chooses which one the ability refers
to at the time the nonactive players become defending players.
All other cases in which the “defending player” is referred to actually refer to both defending players. If the reference involves
a positive comparison (such as asking whether the defending player
controls an Island) or a relative comparison (such as asking whether
you control more creatures than the defending player), it gets only
one answer. This answer is “yes” if either defending player in the
comparison would return a “yes” answer if compared individually. If
the reference involves a negative comparison (such as asking
whether the defending player controls no black permanents), it also
gets only one answer. This answer is “yes” if performing the
analogous positive comparison would return a “no” answer. The same is
true for all other cases that refer to the “attacking player.”