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Jean Scott, co-author of the Tax Help for the Frugal Gambler, answers this one.
The W-2G is issued to the person playing that hand. It doesn't matter whose money it is.
It couldn't possibly be any other way. If the W-2G were based on the money and not the player, anyone could say something like, "Well, I'm playing this session of Let It Ride (or this slot or video poker machine) with money that my girlfriend Lulu back in Indianapolis gave me. So you need to issue her the W-2G. Her address is 123 Main Street. Sorry, but I don't know her Social Security number..."
The issuance of a W-2G to the player also happens to Brad and me at the machines once in a while. Sometimes we switch machines, but forget to switch players cards. If I hit a jackpot on the machine where Brad's card is inserted, the W-2G is issued in my name. Again, if it were issued in the name on the players card, an unscrupulous player could pick up a players card left in any machine and claim that it was John Doe's money, so he should bear the tax burden.