Payments to nonresident aliens are reported to the individual and the IRS on Form 1042-S or the W-2 form, but not on the 1099 form. The W-2 is issued by January 31; the 1042-S by March 15. Payments to resident aliens are reported on the W-2 form or 1099 form, but not on the 1042-S form.
There are no simple answers to this question. Cornell must withhold taxes on all payments to nonresident aliens unless the payment is exempt under the Internal Revenue Code, a tax treaty, or as foreign source income.