NO!! A FE would certainly never be empowered to shut down any engine without a clear command from the handling pilot (or Captain - Company SOPs may vary). Even then, the shut down actions would be monitored by the non-handling pilot (or FO) before the engine was positively shut down and thrust irrevocably lost. NEVER, NEVER would such a critical action be taken without cross-monitoring - and that is one of the cornerstones of CRM!