The Nine and the Three

[Zigûr]

Originally Posted by Inziladun

However, FOTR leaves little ambiguity that Sauron did not have firm knowledge of who kept the Three, though he had suspicions.

I think this is the salient point. Logically speaking, there were only three places where the Three could be kept: Rivendell, Lórien and Mithlond - anywhere in which the Noldor still lingered. Sauron didn't know where the Shire was until Saruman told the Lord of the Nazgûl, which suggests to me that he had not necessarily spied out the West sufficiently to know, for instance, that Círdan no longer possessed one of the Three.
My point is that, according to Sauron's wisdom, to whom would the Three have logically been allocated? Lórien was the closest to both Moria and Dol Guldur, and resistant to assault, which implies a Ring being present at that location. If Sauron knew or at least suspected that Gil-Galad held some of the Rings prior to his death, I think it would make sense, by his logic, that he had passed the greatest of them, Vilya, to one of his subordinates - Elrond or Círdan. That leaves Narya and Nenya to be accounted for.
Despite the fact that he himself did not have a hand in creating them we could also imagine that Sauron was aware of the respective properties of the Three. Depending on the circumstances, this may have led him to at least be able to take an educated guess as to Nenya's location.
I personally don't believe that the bearers of the Nine could detect the location of the Three - it doesn't seem to make sense to me that Sauron could be unaware of their location, yet his lieutenant could. I would venture, personally, that it was more of an accurate supposition deriving from a combination of evidence and guesswork based on his knowledge of the Rings.
Similarly the mention of the White Ring could be an elaboration on the part of the author deriving from superior later knowledge, and so Sauron might have been aware, or could hazard a guess, that one of the Three was in the Golden Wood without being sure which one.
Personally I think that the secrecy surrounding the Three was a last-resort safeguard on the part of their bearers, because I think it must have been completely obvious to Sauron which locations in the West held Elven-rings even if he could not be sure where each individual Ring was located.