In a Two-Headed Giant game, if one player on a team loses the game, that team loses the game.

If you support the Phyrexians, you must choose a target opponent as your end step begins. Whether that player loses the game is determined only as the ability resolves. This means that the card you discard may be the fifteenth artifact card that causes the target player to lose the game, but it also means that if there is no legal target opponent, you won’t draw or discard.

If you support the Mirrans, Mirrodin Besieged’s ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered.

If you somehow control Mirrodin Besieged and no choice was made for it, it has neither of the two triggered abilities.