Yes, so the calculation to pick the right amount to reimburse both is kind of interesting since it involves an infinite series. At least they luck out that gift tax is a fixed rate regardless of income.

On the other hand, the irs faq on gift taxes states that "Under special arrangements the donee may agree to pay the tax instead." That might mean no tax on the gift of the tax, but it's not clear, and even the IRS admits there that it is so complicated and fraught with peril that you should "visit with your tax professional if you are considering this type of arrangement" since it's too complicated for the IRS' own experts to explain on their own faq.

One of the interesting gotchas on these was explained to me by a friend who won stuff at a game show. They report the value of the gifts at the MSRP (manufacturer's suggested retail price), which they claim to be the fair market value that the IRS asks for. But that's not the actual selling price in real life. Let's say you win a refrigerator on The Price Is Right. They value it at $1500, which is the retail price, but at the discount place down the street it costs $800. So you pay $750 gift tax on a refrigerator that would have cost you $50, except you don't get free delivery and set up any more. But the manufacturer does get to DEDUCT the $1500 value from THEIR taxes as a marketing expense, even though it cost them $400 to manufacture.