Both are true. It was his power that made it special (when he created it), and when it was lost from him, it was still imbued with that power. So yes, when Isildur, Gollum, Bilbo, and Frodo bore the Ring we would say it was powerful "on its own", yet that power was from Sauron. But not that he could control it remotely. (And Sauron's lack of the Ring kept him bodily diminished, from having placed so much of himself in an external object, and then becoming separated.)
If by "magical on its own" you mean, did the Ring generate its own new, native power after Sauron lost it, no.