Do I need to file a 1099 for a contractor who lives in a foreign country?

How do I fill out the line "Recipient's ID Number"?

Do I need to withhold US for this contractor, even though she is not subject to US taxes?

If the contractor is a foreign person and works from foreign country, then you do not withhold any tax.

What MukatA says is true, if NO WORK is done within the U.S. borders.

If work is being done IN the U.S. however, you are obligated to withhold 30% of the pay for income tax purposes. If the contractor has no tax ID number, he need to submit Form W-7 to get an Individual Tax Identification Number.