I remember hearing such a play in a radio broadcast involving the Black Hawks long ago, but heck if I can remember who was involved. We were told that one player lost his stick, grabbed a teammates' and scored with it.
This answers the title of the thread though not in conjunction with the intended meaning of the OP. Rick Tocchet, in a scoring slump, experimented in a game using Ed Hospodar's stick and scored twice. There's probably similar stories to that but it is an example of scoring with someone else's stick.