In our company the Captain controlls the power levers on all T/O's. The first pilot to see anything which will affect the safety of flight on T/O prior to V1 will call failure, the Captain calls stopping and does just that. After V1 an engine failure is called "power loss #1/2" to differentiate between stopping before V1 and continuing after. For the FO's T/O he says "call me V1 @ *** set the pwr.", the FO's concentration is on flying the aircraft he/she is not required to gaurd the power levers but to monitor the T/O power set by the Captain.
Checkboard's comment are very relevent re: lawyers. It is not a question of an FO's ability or competance to carry out a RTO but more a legal one. Good SOP's are designed not only to safely fly the A/C but to protect the crew/company as well in the unlikely event.