No, there's only one motor on the (if I remember correctly) left side. It drives both flaps through a small transmission, one side directly to the push rod and the other via cables to the pushrod on that side. The latter is the only part the pilot typically sees.
The real intersting part is the flap handle. It moves two microswitches that track around the flap indicator. If the handle is displaced where the indicator is currently, it actuates either the UP or DOWN microswitches powering the motor.