I tested it myself just now with a song that's lately in my head; namely, Coldplay's "Viva La Vida." I've never had perfect pitch in my life, to my knowledge, so I figured I'd be off by at least a couple notes.
I was exactly on pitch.
My tempo was a little too fast, but I nailed the key. Crazy.
Try it yourself! Tell me what happens.
By the way, it's sometimes hard to remember what notes you were singing, because when you start playing the real song, the legitimate notes wipe clean your memory, at least for most of us. Thus it's difficult to know whether you've matched the pitch or not. I did it by singing a few bars into the computer microphone and saving the file, then playing it back and comparing it to Coldplay's version. I'd overlay them in Sound Studio to demonstrate my moment of accurate pitch, but there's no reason you should have to hear me sing "da da da-da-da-da". (I couldn't remember the lyrics...)