Godhead of Awe's ability overwrites other effects that set a creature's power and toughness to specific values only if those effects existed before the Godhead entered. It will not overwrite effects that modify power or toughness (whether from a static ability, counters, or a resolved spell or ability), nor will it overwrite effects that set power and toughness which come into existence after the Godhead enters. Effects that switch the creature's power and toughness are always applied after any other power or toughness changing effects, including that of the Godhead.

This doesn't cause creatures to lose their abilities.

If two Godheads of Awe are on the battlefield, they'll each make the other one 1/1.