Hi Kate,
I think we know that Sauron put his own power into the One Ring. But then he was the maker of it as well as the wearer. He did it so he could command the other Rings. The Ring did not enhance Sauron's power, except in gaining command of the other Rings. (Of course, losing the Ring diminished his power, because he had put part of his power into it.)
Cirdan, Elrond, Galadriel and Gandalf were not the makers of the Elven Rings. How could those Rings have worked if they could not add anything to the wearer that was not already there? The Elven smiths of Eregion learned how to put power into Rings from Sauron, AKA Annatar, but I think the power they put into the Three Rings was their own power, not Sauron's.
Now, presumably, the Elven Smiths were all Noldor who had once dwelt in the Blessed Realms. If they were looking within themselves for the power to improve understanding, healing and preserving to insert into the Three, could their ideal of those powers be Valinor, and the source of the power that they put into the Three be their memories of Valinor (of both the Seen and the Unseen aspects of Valinor - as we know via Gandalf, or Glorfindel that those who have dwelt in the Blessed Realm have great powers against both)?
If so, could the powers of the Three be fueled by a spiritual connection to the Blessed Realm, and that connection give Elrond a pretty accurate insight into the decisions of the Valar?