So there is likely a place for each of the specialties in placing implants. For the more involved, medically complex patients requiring implants (including zygomatic implants) OMFS is probably the way to go. If you're looking for ideal soft tissue management, probably a periodontist would be the way to go. If you're looking for a holistic treatment planning centered approach and ideal location of implant/prosthetics, then a prosthodontist would be the best way to go. It seems that nowadays the majority of implants are being placed by GP's however.
One thing to keep in mind is that with all the periimplantitis issues we are seeing with implants nowadays, the root cause typically comes down to treatment planning of the implants, not so much the surgery itself. Not sure how much patients will really know about this. With that being said, dentists will refer to with whom they have a better relationship or better treatment outcomes.