The donor is generally responsible for paying the gift tax. Under special arrangements the donee may agree to pay the tax instead.
to be clear, the DONOR is always responsible for any REPORTING or any GIFT TAX.
if the gift is under $17,000, then there is nOTHING for the DONOR to do.
if the gift is OVER $17,000, then the DONOR must REPORT that. There is no tax.
I wouldm't worry about GIFT TAX, that normally only occurs via the estate after the donor passes and is quite unusal under current tax law.
Gifts received are not reported as income on your income tax return.
If the gift is more than $16,000 (2022 limit), the giver (not the recipient) of the gift may be required to file a gift tax return (separate from an income tax return).
"Gift Tax" is somewhat of a misnomer. Even though a gift tax return may be required, very few people ever actually pay federal gift tax. The purpose of the gift tax return is usually only to document a reduction in the allowable estate tax exemption.
See https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Planning-and-Checklists/The-Gift-Tax-Made-Simple/INF12127.html
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.