Telepathy, as most often defined, describes the "transfer of information on thoughts or feelings between individuals by means other than the five classical senses."
I don't believe that any of the examples thus far demonstrate that the elves had a separate "sense" which allowed them to be telepathic. I believe most of the examples of "telepathy" provided in this thread thus far are simply examples of enhanced intuition and perceptivity - examples where the classical "senses" are the only tools necessary for communication - namely sight (and sometimes, touch). Even where Gandalf "strives" with Sauron from afar, it seems he needs to be able to at least "see" the "shadow" in order to guess, and contend with, the mind behind it. There is no need to infer 'telepathy' from those accounts. Even the passage in Many Partings which describes "thoughts" going "to and fro" and individuals looking from "mind to mind" may be interpreted the same way. People who have known each other for thousands of years probably do not need to talk in order to communicate. Heck, my wife and I have known each other for 7 years, and we can say quite a lot to each other with a mere look!
The more important question here, I think, is can the Eldar wordlessly transmit thoughts to each other, and other races, if they cannot see the object of their transmission. If someone can point to a canonical example of such a communion happening over long distances, where the communers cannot see each other, then I would be convinced.
But as a piece of advice to the original posters (if they are still around) I would not take vague hints and "extrapolate" on them for the purposes of fanfiction. Fanfiction often suffers because of this unfortunate derivative practice whereby the original work's wonderful and evocative mystery is made decidedly unmysterious and mundane. If you include 'hints' of what can possibly be interpreated as 'telepathy,' just as the Old Professor did, then I think your fanfiction will be alot richer. Otherwise, it will get lumped in with alot of the other derivative trash (pardon my harshness) that exists out there.