Spanish possessive pronouns regarding gifts?

Suppose that we are close friends and you give me a thoughtful gift. To me, is it "mi regalo" (because it's mine), "tu regalo" (because you gave it to me), or just "el regalo"?

Or, now that I think of it, does it cease to be a gift once it's been given?

Any replies useful, particularly if you can tell me the dialect you're citing.