It is not made clear in LoTR which of the three
eagles picked up the
hobbits.
The paragraph simply reads:
"Side by side they lay; and down swept
Gwaihir, and down came Landroval and Meneldor the swift; and in a dream,
not knowing what fate had befallen them, the wanderers were lifted up and borne far away out of the darkness and the fire."
The best assumption would be that the two unencumbered by
Gandalf picked up the hobbits.
Gandalf specifically asked Gwaihir to bear him.
"‘Twice you have borne
ME, Gwaihir my friend,’ said Gandalf. ‘Thrice shall pay for all, if you are willing."
Then he specifically asked Gwaihir to bring TWO others...presumably for
Sam and
Frodo.
‘Then come, and let your
brother go with us, and some
OTHER (
singular) of your folk who
IS (
singular) most swift!"
Three
birds for three people.
Though it's "possible" that Gandalf may have hoped
Gollum somehow survived, it is doubtful. He had long ago said Gollum may yet have a part to play, in his foresight. It may well be he had at last foreseen what that part would be - else he would likely have asked for yet another bird to come.
A little bit of speculation, but I believe fairly sound given the evidence at hand.
p.s. - it's been awhile since I watched 'the movie' and noticed which eagle actually picks up Frodo... but that part doesn't matter to me. If it's Gwaihir carrying Gandalf, it's all about 'screen time' though it could have been handled differently, more accurately, and with the same 'screen time'.
Gwaihir's only roles in the flight in LoTR are to carry Gandalf, and that it is Gwaihir that first spots the Hobbits when things clear a bit, and begins the spiraling descent.