The parent is required to report, and may owe the tax.
Any person can give any other person a specified amount, now $12,000, each year, without requiring reporting, and there's no reporting on transfers to your spouse.
Ignoring the generation-skipping tax, which doesn't apply on parent-child transfers, amounts over that are reported by the giver on form 709. There's no tax due on that until the total gifts FROM that person reach $1,000,000. Amounts over that $1,000,000 have gift tax due on April 15 of the following year. If gift tax is paid, the recipient needs a copy of the 706, as part of the gift tax paid adds to the basis of the property transferred.
The following is not exactly correct, but is a reasonable approximation. The untaxed reported gifts are added to the estate for the purpose of the estate tax on the giver's death. After the exemption (now $2 million) is applied, the rest of the estate is subject to estate tax on form 706. (It is important for the filed 706 forms, or at least the most recent one, to be made available to the executor of the estate so that the proper numbers can be transferred to the estate tax return.)What is the gift tax owed if a parent gives a home to a child?
The parent fills out the form 709. The form keeps track of lifetime taxable gifts. (Even if the $1Million is not exceeded, the total counts against the estate.)
The child needs a copy of the gift tax form since some of the calculated gift tax is added to the basis (which starts at the parents basis--the capital gains are passed along to the child).
The giver owes the tax if any is due, not the recipient.
If the gift is over $12,000, the giver must file a gift tax return. But they won't pay a gift tax unless they have already used up their $1 million lifetime exemption.