Is it common during a FUE procedure for technicians to do the some/all of the implanting?

The answer is no, it is uncommon for physicians to place grafts, especially all of them.  Most facilities rely on technicians to hand place grafts in premed surgical slits completed by the physician.
In my office, I core and place all of the grafts on every patient.  I use implanter pens that are loaded by one or two assistants and placed by me, using a stick and place technique with no pre-made surgical slits.  I feel that the healing and results are superior. My patients have a tremendous comfort level knowing that I will be placing each graft and that I am fully involved throughout the entire case.  In doing so,I am able to ensure that each graft placement is done in the correct manner and with the least amount of trauma. 
I encourage all prospective patients to interview their physicians to determine their involvement in the case to gain the greatest comfort level prior to making a decision on who performs it.

In my clinic I do all - extraction and implantation as well. I use implanters and technicians only load the tools. I know in other clinics technicians insert follicles and it is common.