DDD Study - Requirements For Paying Foreigners With Visas:

J-1 and J-2: Documentation of authorization by the Immigration and Naturalization Service for employment.

F-1: Documentation that the following requirements have been met:

a) The student has been in F-1 status for one full academic year.
b) The student MUST request a recommendation from a Designated School Official for off-campus employment.
c) The off-campus location is educationally affiliated with the F-1 holder's school. The employment MUST be an integral part of the student's educational program. 
d) Employment MUST NOT EXCEED 20 HOURS PER WEEK while school is in session. During holidays or school vacation, an F-1 holder may work full-time.
e) The student has demonstrated that employment will not interfere with the student's carrying of a full course load.

H-Visas: can only receive payment from the employer sponsoring the visa.