Technically, if gift tax is owed, and if the giver does not pay it, the IRS may come after the recipient for the taxes.
However, as stated above, under US law, gifts are not even reportable by the giver unless the given amount is more than $16,000 per person for 2022 or more than $17,000 per person per year for 2023. Even if the gift is larger and must be reported, gift tax is not owed by the giver unless their lifetime total of gifts and estate is more than $13 million.
So even though there is a rare technical circumstance where a recipient might be asked to pay gift tax, it almost never actually happens.
Separately, although you did not mention this as a factor, if you received a gift from a foreign person of more than $100,000 (including family) or from a foreign company of more than $17,000, you must report the transfer, even though it is not taxable.
