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The short answer is yes. According to the IRS, if you're an American taxpayer and you earn money anywhere in the world, you must declare and pay taxes on it.
However, nowhere on the tax forms are you required to specify what kind of gambling you're doing or where you're doing it. Your wins and losses are simply enumerated under the catchall category of "gambling."
Of course, offshore Web-based gambling sites don't issue you W-2Gs or 1099s; they don't even ask for your Social Security number. Some gamblers surely don't claim offshore gambling winnings; some gamblers who win money in domestic casinos and manage to fly under the tax-doc radar also don't. But you take your chances if you don't.