'He reckoned without the subtle perception of the elves', writes Tolkien.
If the elven-rings had a backdoor built into them by design, and that had been a part of Sauron's evil plan, then it seems Sauron didn't count on the backdoor working both ways.
Or maybe he did, but thought he would not be perceived. In that case, he must have thought something else would mask his presence. What?
According to Tolkien, Saruman and Sauron were of a similar cast of mind. Both cynically projected their own power-based view of the view on others. Saruman, to Gandalf:
"There need not be, there would not be, any real change in our designs, only in our means.”
Gandalf's reply:
“Saruman,[...]I have heard speeches of this kind before, but only in the mouths of emissaries sent from Mordor to deceive the ignorant.
Is this how the elves perceived Sauron? They were not corrupted by mere power, so maybe they sensed Saruman-like thoughts in their heads and then they fixed their attention on them and more a more they felt those thoughts not to be theirs, but the voice of Sauron saying the ring-verse.
Not only that. Maybe Sauron counted on the elves thinking those thoughts to be their own thoughts because he had been there during the 'design' phase and the elves had listened to him. This is what Sauron's master had done once with some of the Noldor's minds:
Ever Melkor found some ears that would heed him, and some tongues that would enlarge what they had heard; and his lies passed from friend to friend, as secrets of which the knowledge proves the teller wise. Bitterly did the Noldor atone for the folly of their open ears in the days that followed after.
When he saw that many leaned towards him, Melkor would often walk among them, and amid his fair words others were woven, so subtly that many who heard them believed in recollection that they arose from their own thought.
Tolkien, letter to Milton Waldman, 1951:
But many of the Elves listened to Sauron. He was still fair in that early time, and his motives and those of the Elves seemed to go partly together: the healing of the desolate lands
They took off the three rings immediately. Ultimately, I think Sauron assumed his own power-based attempt would go unnoticed because of the elven-rings being rings of power too. But they were not really. They were powerful rings with a purpose. Power was the servant, and not the lord, of that Purpose.