Footnote 7 to the History of Galadriel & Celeborn (in Unfinished Tales) has some interesting comments to make on this. This footnote discusses what CT calls "an isolated and undateable note" and first of all mentions the fact that the Elves were not even aware that Sauron was still active until about SA 1600 and the time of the forging of the One Ring.
It then goes on to discuss Annatar:
When he came among the Noldor he adopted a specious fair form (a kind of simulated anticipation of the later Istari) and a fair name: Artano "high-smith," or Aulendil, meaning one who is devoted to the service of the Vala Aulë. The note goes on to say that Galadriel was not deceived, saying that this Aulendil was not in the train of Aulë in Valinor, "but this is not decisive, since Aulë existed before the 'Building of Arda,' and the probability is that Sauron was in fact one of the Aulëan Maiar, corrupted 'before Arda began' by Melkor."
This is, of course, typical Tolkien, providing a variant version of things (in this case the name used by Sauron, although there's no reason why he had to restrict himself to a single name), but the key point here is that in this note at least Tolkien acknowledged that Sauron would have openly declared himself as a servant of Aulë.
Earlier in the same text, we read about Sauron's arrival among the Elves:
When he felt himself to be secure he sent emissaries to Eriador, and finally, in about the year 1200 of the Second Age, came himself, wearing the fairest form that he could contrive.
In Eregion Sauron posed as an emissary of the Valar, sent by them to Middle-earth ("thus anticipating the Istari") or ordered by them to remain there to give aid to the Elves.
Based on this the Elves, of course, would have thought he was a servant of Aulë, although Galadriel's (inconclusive) suspicion is noteworthy.
While it would be interesting to speculate about the emissaries he sent, I'm not aware of any other mention of them in the texts.
Further on the text mentions that at some time between SA 1350 and SA 1400 Sauron had persuaded the Elves of Eregion to rebel against Galadriel and Celeborn, and that he had departed from Eregion by SA 1500, shortly after work had begun on making the Rings of Power.
His time there was approx. 300 years, but by about hafway through that time he had already won the trust of the Elves sufficiently enough to provoke them to a rebellion.