The mental communication between Galadriel, Elrond and Gandalf was most definitely not due to the Rings - their power was explicitly lost when the One was destroyed.

For the answer to that one we must look to the "Quendi and Eldar" essay published in the "War of the Jewels", section headed "Note on the language of the Valar":

For the Valar and Maiar could transmit and recieve thought directly

From there a CJRT footnote directs us to Annals of Aman paragraph 164, and from there his commentary references the RotK scene.

Gandalf/Olorin, as a Maia, is quite capable of recieving thought from the Elves and transmitting thought to them (this is touched on in Q&E in the paragraph following the quote I give), so even if the Elves can't do so between each other, Gandalf/Olorin certainly can facilitate it (much as a translator can facilitate communication between two people who don't speak a common language).

So much for the effect of the Rings here; back to the main question. This is answered in "Of the Rings of Power and the Third Age" (pulished in the Silmarillion) and I quote in full:

Therefore ere the Third Age was ended the Elves perceived that the Ring of Sapphire was with Elrond, in the fair valley of Rivendell, upon whose house the stars of heaven most brightly shone; whereas the Ring of Adamant was in the Land of Lorien where dwelt the Lady Galadriel. A queen she was of the woodland Elves, the wife of Celeborn of Doriath, yet she herself was of the Noldor and remembered the Day before days in Valinor, and she was the mightiest and fairest of all the Elves that remained in Middle-earth. But the Red Ring remained hidden until the end, and none save Elrond and Galadriel and Cirdan knew to whom it had been committed.

So, your answer is: "yes".

The mental communication between Galadriel, Elrond and Gandalf was most definitely not due to the Rings - their power was explicitly lost when the One was destroyed.

For the answer to that one we must look to the "Quendi and Eldar" essay published in the "War of the Jewels", section headed "Note on the language of the Valar":

For the Valar and Maiar could transmit and receive thought directly

From there a CJRT footnote directs us to "The Annals of Aman," paragraph 164, and from there his commentary references the Return of the King scene.

Gandalf/Olórin, as a Maia, is quite capable of receiving thought from the Elves and transmitting thought to them (this is touched on in Q&E in the paragraph following the quote I give), so even if the Elves can't do so between each other, Gandalf/Olorin certainly can facilitate it (much as a translator can facilitate communication between two people who don't speak a common language).

So much for the effect of the Rings here; back to the main question. This is answered in "Of the Rings of Power and the Third Age" (published in The Silmarillion) and I quote in full:

Therefore ere the Third Age was ended the Elves perceived that the Ring of Sapphire was with Elrond, in the fair valley of Rivendell, upon whose house the stars of heaven most brightly shone; whereas the Ring of Adamant was in the Land of Lórien where dwelt the Lady Galadriel. A queen she was of the woodland Elves, the wife of Celeborn of Doriath, yet she herself was of the Noldor and remembered the Day before days in Valinor, and she was the mightiest and fairest of all the Elves that remained in Middle-earth. But the Red Ring remained hidden until the end, and none save Elrond and Galadriel and Círdan knew to whom it had been committed.

So, your answer is: "yes".