TL;DR: apparentlyAt least for the same as other Elvish crafts that appear 'magical' to mortalsThree, not Sauron's powerand probably the Seven and Nine: from the skill of the Elves that made them
(Invisibility EDIT to add more about the Seven and Nine in response to the OP's request in comments:
From the same letter:
And finally they had other powers, more directly derived from Sauron ('the Necromancer': so he is called as he casts a fleeting shadow and presage on the pages of 'The Hobbit'): such as rendering invisible the material body, and making things of the invisible world visible.
Unfortunately it is a powernot entirely clear what "more directly derived from Sauron" permeans - do the same letterSeven and Nine actually contain a bit of his being, or does it just mean that the powers are Sauron's design, or derived from Sauron's magic but not his personal essence? The reference to his title as "the Necromancer" may suggest the latter - that the invisibility/wraith-transformation powers come from Sauron's evil magic - but it is not stated exactly.)
Of the Third Age and the Rings of Power states that the Three were the most powerful of the Elven-made Rings - "Now these were the Three that had last been made, and they possessed the greatest powers" - which may also argue against the Seven and Nine containing Sauron's actual being/power.