NSA cannot request GCHQ to (illegally) spy on you. If the NSA acquires legal grounds to surveil you, such as through the FISC, they may. Although, given that a US citizen is much more likely to use US based service providers, they would likely work with the FBI to obtain your communications than go overseas. http://www.dtic.mil/whs/directives/corres/pdf/524001r.pdf
> C1.1.4. DoD intelligence components shall not request any person or entity to undertake any activity forbidden by Executive Order 12333 (reference (a)).
Similarly, GCHQ cannot ask NSA to illegally spy on you. http://www.legislation.gov.uk/ukpga/2000/23/section/1
> it shall be the duty of the Secretary of State to secure that no request for assistance in accordance with the agreement is made on behalf of a person in the United Kingdom to the competent authorities of a country or territory outside the United Kingdom except with lawful authority.
And of course you may assume that such agencies violate these governances, but that would show a lack of understanding of the bureaucratic nature of such agencies.