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As far as I know the rings didn't grant any kind of communicative ability. That would be cool if they did though. The power of Galadriel's ring was pretty much consumed by maintaining Lothlorian - once the one ring is gone Lothlorien ceases to exist. The last of the two way connections are the palantirs - but only those with the strongest wills dare touch them.

The high elves, and Gandalf, certainly had much ability to communicate from long distances... see things that happen and perhaps send messages. They are mysterious that way. Galadriel sensed that Gandalf was not really dead somehow, certainly, and sent for him at Caradhras.

How many times in the story were messages sent on ahead by the elves, in mysterious ways we didn't really understand? The elves the the hobbits met in the Shire sent messages that reached Rivendell, and Galadriel knew the Fellowship had left Rivendell, even though no messengers could have beat the Fellowship there when they arrived... It is one of the mysteries PJ was interested in enough to show in the film.