[Mithadan]

Regarding the Silmarils, I think they were "powered" by the light of the two Trees. I do not have access to my books, but I seem to recall that in HoME, it is stated that Elvish "magic" is a function of their own nature and inherent power, so the act of creation requires that some of the maker's essence be poured into the item created. But their light had an outside source.

Regarding the Three Rings, and presumably the Seven and the Nine, some portion of their makers' essence would have been instilled into their creations. Some of Sauron's power would have been placed in the Seven and the Nine as he assisted in their creation. Some of the power of Celebrimbor and his smiths would have been placed in the Three. But the power of the Three to act may have been drawn from other sources. Pitchwife discusses their nature above and their links to the elements, air, water and fire. Their power may come from those elements to some extent.

The role of Elves in Middle Earth was, in part to heal the damage caused by Morgoth and do what they could to fix Arda Marred. They are described as being far more linked to nature and their surroundings than Men. This may have been part of the nature and purpose of the Three. Healing, preservation and staying the effects of time on their wearers and their people.

Has anyone ever considered the nature and intricacy of Sauron's plot? He spent decades or centuries instructing the Elves of Hollin and assisting them in the making of the Great Rings, with the intent to create the One in order to dominate the bearers of the various Rings and their people.