in the US, they can't. I don't think the FAA and PAX would be very happy to have "student"pilots in the cockpit.
in Europe, no rules...they could even ask a captain to pay if he want keep his job.as captains are no so idiot, they ask copilot to pay for everything.
as long it is authorized, why shouldn't they???they could even ask you to pay a participation in the aircraft.
(I know it is not authorized by the french DGAC)