Does a fighter pilot who flies the F/A-18E have permission to fly the F/A-18F for training a new pilot? Or does an F/A-18E pilot fly with an F/A-18F for night bombardment?
Typically yes, but they generally won't be training a new pilot one day and flying a night bombing sortie the next.
Two seat trainers are intentionally very similar to their single seat counterparts. (for obvious reasons!) A pilot training in a squadron to replace front line fleet F/A-18E pilots will likely conduct early stage familiarization flights in the F model, but later solos will be in the E model. And once training is complete the operational squadron he/she is assigned to will only fly the E model.
After an operational tour, if a pilot is assigned to an instructor tour there might be a refamiliarization flight in the F model as part of the IUT (Instructor Under Training) syllabus, plus a written test on any procedural or limitations differences. Since the front seat of the two seat trainer is in the same location as the seat in the single seat model, the student will sit in front for most flights, with the instructor in back. The IUT syllabus would emphasize the change in perspective when flying formation, and performing or observing other maneuvers while sitting in the rear seat.
In summary, a pilot would be cleared to fly either model as part of the training process for being designated as a qualified instructor.
They then might fly the F model for early stage flights, and the E model as formation lead for flights later in the syllabus. But they wouldn't go on a night bombing mission as a shore-based instructor, unless of course it was a training flight.