Not just the rancher, usually the farmhands. Usually they are very, very well trained, usually by whoever was the most knowledgeable person before. It's kind of like passing down Grandma's apple pie recipe over the generations. They know how to handle the guns, how to handle the tanks , the semen, and get a "feel" or exactly where to put it, just as good as a vets ...something that only comes with experience. Seriously, you may laugh at the term magic inseminators, but some of these guys have rates that far exceed vets.
Of course, sometimes it gets muddled over the course of years and you'll see a rise in services per conception, and then the semen distributor rep vet comes out and figures what you're doing wrong, fixes it, etc.
But the vast majority of the time, you have people that at least competent and usually quite skilled. Any training for new people takes place under the guidance of the "old hands", the semen distributor reps, or a vet.
It is *very* common esp., in high producing areas, for staff to take care of injections, heat detection, insemination, the manual preg checks, a lot. Vet comes out every six months or so to check records, or comes over if there is a problem. It just isn't practical to ask the vet to come out day after day to administer all the injections and inseminate tons of cattle and check *all* of the pregnancies at at all sorts of dairies. If it's a small dairy and he has nothing else to do, sure. But few dairymen are going to pay for that, so they train their staff.
That is why, if there is a repro problem on the farm, as a vet you need to TALK TO THE DAIRYMEN/HERDSMEN and not just the owner, because they are involved SO much with the animals.