Here's the thing. Magic isn't some thing that you "use" in Middle-earth. It's not like the Force or whatever it is that they draw from in the Wheel of Time books. It doesn't exist on its own.
What some people perceive as magic is really just the inherent nature of certain entities that exist within Middle-earth. The Wizards belong to the race of the Ainur which are basically like angels. They don't "use" magic. Whatever power they display is simply a part of their nature as Ainur. To races that don't have the same nature, it seems magical but only because that power is something the other races do not have as part of their nature. Men cannot do the things that the Istari can do and they cannot quite understand it, so they call it magic.
Elves are the only other inherently "magical" race in Middle-earth. And Tolkien actually goes to the trouble in some of his writings (published posthumously in the History of Middle-earth series) to explain exactly what it is about the nature of Elves that makes them seem magical to the races that are not like them. Elves are unique. And Tolkien states that they are the only "magical" race in Middle-earth.
So, to give you the short answer to your question, Elves are the only characters that I would accept acting in "magical" ways. Anyone from another race trying to pass themselves off as having "magical" abilities I would have to assume is either delusional or working for Sauron. (Since the only other way we know of to gain such abilities besides partaking in one of the natures that Eru created as having such super human power is to have been given such power by the Dark Lord.)
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