It is true that the casino can check all of their members and they do that too for security purposes. But while it is a matter of tax I think gambling sites should bear the tax of their end which they earn only. And for the users, the users are liable to their country whether they will pay or not.
There are two type of taxes that the casino will have to remit.
1- income tax.. that is their income tax due from their operation, that could be quarterly, semi annual but mostly it's annually.
2- witholding tax ... this is the tax that the casino witheld from the gamblers that wins a significant amount of money, this method is employed to ensure that taxes are remitted as the government doesn't trust the gamblers to declare their winnings.