I really grateful for this deep and witty answers. I also thought about Mouth Of Sauron's words to Gandalf but could not remember all these passages from other literature.
As Istari did not disclose their nature, it definitely took some time for Sauron to start considering their connection to the West. At some point upon arrival Istari had to reveal some extraordinary capacities - otherwise no-one would have paid any attention to old men; they had to demonstrate that they were wizards. At least worthy for fireworks and other special effects.
Although they could have come to Sauron's (or Nazguls') attention quite early, it was not before they had made some considerable headache to Sauron and his forces, he started connecting them to the West.
I wonder if Sauron was ever able to establish he was facing other Maiar. Don't think, Saruman was eager to let Sauron know, even when he came under Sauron's control. This control was not a kind of domination like Sauron enjoyed over Nazgul, for instance, and Saruman (in the book) continued his search for the Ring, his own game.
But if Sauron had known, would he have done something in a different way? For example, knowing there is a Maya in Gondor (who had just managed to have a Balrog killed), would Sauron have sent the host under Witch King to Minas Tirith? Or would he have rather waited for more reinforcements? Or the situation with the Ring seemed so pressing that Sauron, in fact, had little choice?
And if Sauron knew or suspected Gandalf was a Maia, what should he have thought on the fact that Gandalf did not claim the Ring that was in his proximity for so long, especially when he was safely beyond Sauron's reach?