The employer-employee relationship is determined by the nature of the work, not the forms you use. If you control the person's working conditions and hours, they are an employee and must receive a W-2, and you have all the other duties that you would have for any other employee. They are not independent contractors just because you can't or don't want to issue a W-2
However, an ITIN is not valid for a W-2. An ITIN is a tax number that is only available to people who are not eligible for social security numbers, and do not authorize a person to work in the US. If the employee is authorized to work in the US, they should apply for an SSN and you should use that.
If the employee is not authorized to work in the US but you want to employ them anyway, you probably need legal advice. Paying them on a 1099 is incorrect, paying them on a W-2 is incorrect, and paying them "under the table" is definitely incorrect. Seek the advice of a tax attorney or immigration attorney.
If you are paying a non-citizen who lives abroad (non-resident alien) for services that are performed wholly abroad, then that appears to be non-US income to them that is not subject to any US tax laws. (It may be subject to tax in their home country. See here https://www.irs.gov/individuals/international-taxpayers/persons-employed-abroad-by-a-us-person) Form 1042-S is used to report certain US payments to foreign persons but not wages, as best I can tell. You may want to seek professional guidance.
