Like SpM I'm sure that the locations of both Lothlorien and Rivendell would have been known to Sauron. How did he find out though? It is possible that he discovered this through powers of osanwe or thought- transference, if you've not read the threads located
here and
here , and I wouldn't be shocked if you hadn't as they are quite long and 'deep'
. Although, the Three Rings pre-date the creation of these havens, so it could be down to simple espionage.
There are a few interesting things about Saruman and the nature of his treachery. he was clearly using the Palantir for his own nefarious purposes, which led inadvertently to Sauron gaining access to his mind and being able, to a certain extent, to control Saruman. But, it is also clear that Sauron did
not have full control over Saruman, and I often wonder who was tricking who? And I also wonder whether in some way his 'power' was equal to or greater than that of Sauron.
Saruman carried out some very clever double-dealing by agreeing with the White Council in driving Sauron out of Dol Guldur so he could carry out his own search for the One Ring. I think this is the essence of his treachery; that he could betray the White Council by deceiving them as to his motives and also by keeping information from them. He deceived
both of those he 'served' in pursuit of his own heretical aims, which not only shows the depth of his knowledge and skill, but how far he could sink in moral terms.