No. Gift tax applies to the gift-giver (donor) not the receiver (donee). The person who gave the gift will have to receive tax advice and probably be told to file a gift tax return to report the gift, although they may not have to pay any gift tax now.
This is not legal advice nor intended to create an attorney-client relationship. The information provided here is informational in nature only. This attorney may not be licensed in the jurisdiction which you have a question about so the answer could be only general in nature.
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If you receive a monetary gift you will not have to pay any taxes as the recipient. The donor can benefit from your gift because of (1) reduced estate taxes and (2) reduced income taxes. Moving money out of an estate via lifetime gifts removes future appreciation on the asset. Plus, if you give property that has a low tax basis (such as a rental house that has depreciated way below its fair market value), or property that generates a lot of taxable income, you may reduce income taxes paid within a family by shifting these assets to family members in lower tax brackets.
My comments are NOT LEGAL ADVICE. They are for informational purposes only. Actual legal advice can only be provided after you have signed an engagement letter. Answering this question does not create an attorney client relationship. Remember that without attorney client privilege you could possibly divulge information that can hurt your legal rights in the future. I am a tax attorney in Miami Florida. I can help you with your federal tax issues via a secure client portal if required.