They were statues possibly made by the Numenoreans of Gondor and inhabited by spirits of Sauron
As far as I can currently tell you, there is no explicit statement by Tolkien of what they were or their origins. However, we can draw some things conclusions about them from the passages they're mentioned in.
There first mention in the Lord of the Rings comes from Gollum suggesting a different route to the Black Gate when Frodo states that he intends to enter Mordor.
'Well, master, there it was and there it is: the tall tower and the white houses and the wall; but not nice now, not beautiful. He conquered it long ago. It is a very terrible place now. Travellers shiver when they see it, they creep out of sight, they avoid its shadow. But master will have to go that way. That is the only other way, for the mountains are lower there, and the old road goes up and up, until it reaches a dark pass at the top, and then it goes down, down, again – to Gorgoroth.' His voice sank to a whisper and he shuddered.
'But how will that help us? ‘ asked Sam. 'Surely the Enemy knows all about his own mountains, and that road will be guarded as close as this? The tower isn’t empty, is it?'
'O no, not empty!' whispered Gollum. 'It seems empty, but it isn’t, O no! Very dreadful things live there. Orcs. yes always Orcs; but worse things, worse things live there too. The road climbs right under the shadow of the walls and passes the gate. Nothing moves on the road that they don’t know about. The things inside know: the Silent Watchers.'
The Two Towers, Book IV: Chapter 3 - The Black Gate is Closed
In their first mention, Gollum seems to suggest sentience for the watchers, providing the first indication that they may be inhabited by some form of evil spirits, suggesting the watchers "live there".
Their sentience is further supported when the watchers seem to be able to feel the power of the Ring as it gets closer.
" 'Bad business,' said Gorbag. 'See here -- our Silent Watchers were uneasy more than two days ago ...' "
Return of the King, Book VI: Chapter 1 - The Tower of Cirith Ungol
The watchers appear to be able to sense the coming of the Ring and are trying to warn Gorbag's orcs of it's arrival. Although Gorbag's orcs aren't due out for another day and without confirmation from Lugurz don't head out.
Sam's struggles with the gates seem to be the most revealing. His description of the gates seem to suggest they are a creation of evil. With their vulture-like heads being something that one wouldn't expect the faithful of Numenor to have built.
‘Come on, you miserable sluggard!’ Sam cried to himself. `Now for it!’ He drew Sting and ran towards the open gate. But just as he was about to pass under its great arch he felt a shock: as if he had run into some web like Shelob’s, only invisible. He could see no obstacle, but something too strong for his will to overcome barred the way. He looked about, and then within the shadow of the gate he saw the Two Watchers.
They were like great figures seated upon thrones. Each had three joined bodies, and three heads facing outward, and inward, and across the gateway. The heads had vulture-faces, and on their great knees were laid clawlike hands. They seemed to be carved out of huge blocks of stone, immovable, and yet they were aware: some dreadful spirit of evil vigilance abode in them. They knew an enemy. Visible or invisible none could pass unheeded. They would forbid his entry, or his escape.
Hardening his will Sam thrust forward once again, and halted with a jerk, staggering as if from a blow upon his breast and head. Then greatly daring, because he could think of nothing else to do, answering a sudden thought that came to him, he drew slowly out the phial of Galadriel and held it up. Its white light quickened swiftly, and the shadows under the dark arch fled. The monstrous Watchers sat there cold and still, revealed in all their hideous shape. For a moment Sam caught a glitter in the black stones of their eyes, the very malice of which made him quail; but slowly he felt their will waver and crumble into fear.
He sprang past them; but even as he did so, thrusting the phial back into his bosom, he was aware, as plainly as if a bar of steel had snapped to behind him, that their vigilance was renewed. And from those evil heads there came a high shrill cry that echoed in the towering walls before him. Far up above, like an answering signal, a harsh bell clanged a single stroke.
These paragraph gives us quite a lot to look at and learn about these watchers. In the initial paragraph we learn that Sam experiences what seems to be a physical barrier, although an invisible one. Sam seems to experience a test of power (which occurs at various times throughout Tolkien's Legendarium, such as with the Palantiri) which he proves to be too weak-willed for.
The Watchers are later described as again being sentient, or inhabited by some sort of evil spirit, which (as we know from the Silmarillion) Sauron may have trapped in the stone structures. The spirits seem to flee from the light of the phial and have a "glitter in the black stones of their eyes". In the final paragraph of the excerpt as soon as Sam returns the phial to a pocket they return to their sentries and are able to give out a "shrill cry".
The look of the watchers is inherently evil. With "vulture-faces" and "claw-like hands". Described as being "immovable and yet aware" again supports the idea that some form of evil spirits were trapped in the stones by Sauron or the necromancy of the Witch-King (which is far less likely). This seems to be the main evidence countering the idea that these statues weren't creations of the Numenoreans, however it is entirely possibly that the original watchers of Cirith Ungol were altered later by Sauron and spirits forced to inhabit them.
After passing through the gate and going up the tower, Sam overhears Snaga and Shagrat discussing the cry and the presence of Sam. This is where they seem to suggest that the watchers were indeed the work of the "tarks", which were the Numenoreans of Gondor.
`Well, you put his back up, being so high and mighty. And he had more sense than you anyway. He told you more than once that the most dangerous of these spies was still loose, and you wouldn’t listen. And you won’t listen now. Gorbag was right, I tell you. There’s a great fighter about, one of those bloody-handed Elves, or one of the filthy tarks. He’s coming here, I tell you. You heard the bell. He’s got past the Watchers, and that’s tark’s work.
The orcs seem to be convinced that only a "great fighter" could get passed the Watchers, including the tarks as that is a work of there's.
On the way out, we again experience this challenge of power from the watchers
At length they came to the door upon the outer court, and they halted. Even from where they stood they felt the malice of the Watchers beating on them, black silent shapes on either side of the gate through which the glare of Mordor dimly showed. As they threaded their way among the hideous bodies of the orcs each step became more difficult. Before they even reached the archway they were brought to a stand. To move an inch further was a pain and weariness to will and limb.
Frodo and Sam are physically prevented from moving forward by another power, which seems to be testing the strength of their will and determining them unworthy. Furthermore, even before that, they feel the Watcher's gaze "beating on them", testing their strength and will.
The strength of the watchers seems to have been broken later by the power of Galadriel and her phial.
The will of the Watchers was broken with a suddenness like the snapping of a cord, and Frodo and Sam stumbled forward. Then they ran. Through the gate and past the great seated figures with their glittering eyes. There was a crack. The keystone of the arch crashed almost on their heels, and the wall above crumbled, and fell in ruin. Only by a hair did they escape. A bell clanged; and from the Watchers there went up a high and dreadful wail. Far up above in the darkness it was answered. Out of the black sky there came dropping like a bolt a winged shape, rending the clouds with a ghastly shriek.
The destruction of the Watchers after the second passing of Sam seems to have summoned a Nazgul, this suggest the two may have been tied in some way, and the Nazgul was made aware of the passing of Sam and Frodo and the destruction of the Watchers. This is the first time we get a mention of the watcher's will, but it possessing one seems to suggest some form of spirit or sentient being having been trapped in the rock.
The Watchers are highly suggested as being sentient and are likely evil spirits trapped in the rocks by Sauron, or another Necromancer. Whether or not their origin lies with the tower itself and the Numenoreans of Gondor, or after the fall of the tower and the armies of Sauron is debatable, yet I believe the evidence provided by Shagrat outweighs the counter evidence. It is clear Sauron had some influence on the Watchers due to their evil heads and hands.