Tolkien has said at various times that the Eagles were serving as agents of Manwë and/or Eru, and he has also showed them being loyal to a few individuals, namely Turgon, Gandalf, Radagast, and Galadriel.
The fëar of Elves and Men and still later things (Ents? Dwarves) were intrusions by Eru, like the Valar – Aule and the Dwarves. Yavanna and the Ents. Maiar could take forms of Eagles etc. – [?these] were sent into Eä. They are not of Eä, but Eru’s agent in Eä.
The Nature of Middle-earth - The Primal Impulse (c.1959)
But Manwë Súlimo, highest and holiest of the Valar, sat upon the borders of the West, forsaking not in his thought the Outer Lands. For his throne was set in majesty upon the pinnacle of Taniquetil, which was the highest of the mountains of the world, standing upon the margin of the Seas. Spirits in the shape of hawks and eagles flew ever to and from his halls; and their eyes could see to the depths of the sea and could pierce the hidden caverns under the world, and their wings could bear them through the three regions of the firmament beyond the lights of heaven to the edge of Darkness. Thus they brought word to him of well nigh all that passed in Aman: yet some things were hidden even from the eyes of Manwë and the servants of Manwë, for where Melkor sat in his dark thought impenetrable shadows lay.
Morgoth's Ring - Ainulindalë (c.1951)
Manwë however sent Maia spirits in Eagle form to dwell near Thangorodrim and keep watch on all that Melkor did and assist the Noldor in extreme cases.
Morgoth's Ring - The Annals of Aman (c.1958)
Then Manwë awoke, and he went down to Yavanna upon Ezellohar, and he sat beside her beneath the Two Trees. And Manwë said: ... But do you not now remember, Kementári, that your thought sang not always alone? Did not your thought and mine meet also, so that we took wing together like great birds that soar above the clouds? That also shall come to be by the heed of Ilúvatar, and before the Children awake there shall go forth with wings like the wind the Eagles of the Lords of the West.'
The War of the Jewels - Of the Ents and the Eagles (c.1963)
But it is not said that Manwë abandoned them, ... His messengers could come from Valinor and did so, and though in disguised form and issuing no commands, they intervened in certain desperate events.
[Footnote:] The most notable were those Maiar who took the form of the mighty speaking eagles that we hear of in the legends of the war of the Ñoldor against Melkor
The Nature of Middle-earth - Manwë's Ban (c.1972-3)
But when he had dismissed Sorontar, Turgon sat long in thought, and he was troubled, remembering the deeds of Húrin. And he opened his heart, and he sent to the Eagles to seek for Húrin, and to bring him, if they could, to Gondolin. But it was too late, and they saw him never again in light or in shadow.
The War of the Jewels - The Wanderings of Húrin (late 1950s)
It seemed that Bilbo was not going to be eaten after all. The wizard and the eagle-lord appeared to know one another slightly, and even to be on friendly terms. As a matter of fact Gandalf, who had often been in the mountains, had once rendered a service to the eagles and healed their lord from an arrow-wound. So you see 'prisoners' had meant 'prisoners rescued from the goblins' only, and not captives of the eagles. As Bilbo listened to the talk of Gandalf he realized that at last they were going to escape really and truly from the dreadful mountains. He was discussing plans with the Great Eagle for carrying the dwarves and himself and Bilbo far away and setting them down well on their journey across the plains below.
The Hobbit - Chapter 6 - "Out of the Frying-Pan into the Fire"
`That was the undoing of Saruman's plot. For Radagast knew no reason why he should not do as I asked; and he rode away towards Mirkwood where he had many friends of old. And the Eagles of the Mountains went far and wide, and they saw many things: the gathering of wolves and the mustering of Orcs; and the Nine Riders going hither and thither in the lands; and they heard news of the escape of Gollum. And they sent a messenger to bring these tidings to me.
The Lord of the Rings - Book II Chapter 2 - "The Council of Elrond"
'Do not let me fall!' I gasped, for I felt life in me again. 'Bear me to Lothlórien!'
'That indeed is the command of the Lady Galadriel who sent me to look for you,' he answered.
The Lord of the Rings - Book III Chapter 5 - "The White Rider"