Unarmed Strike
At 1st level, a monk gains Improved Unarmed Strike as a bonus feat. A monk’s attacks may be with either fist interchangeably or even from elbows, knees, and feet. This means that a monk may even make unarmed strikes with her hands full. There is no such thing as an off-hand attack for a monk striking unarmed. A monk may thus apply her full Strength bonus on damage rolls for all her unarmed strikes.
As per described, it seems for monks (and probably for monks only), using two-weapon-fighting would allow the full STR Adj. to be applied to all of his attacks (flurry of blows, normal main hand attack, off-hand additional attack). Is that how it works?
Also, it seems this feature does not apply to a fighter who picks Improved Unarmed Strike as a bonus feat, but Power Attack specifically allows it to be applied to unarmed attack. So the damage bonus from Power Attack could be usable for a fighter who use unarmed attack for off-hand (though STR Adj. would still be halved)?