No. Gifts are not taxed to the recipient, regardless of amount or whether they are from people in or outside the United States. If you make gifts, you need to check on whether you must pay any gift tax (not income tax) and/or whether you have to file a gift tax return, Form 709.
The tax on the investment income in excess of $1,700 (may change after 2007) of a child under age 18, based on the parents’ marginal tax rate and computed on Form 8615.