There will always be people around who can train dogs. The military centralizes training for standardization & because that's the military way, but the skill of training a working dog is not so arcane that it will die out. It is, however, definitely a skill that has to be learned, and the work of training dogs to a high level is going to continue to be specialized.
Dogs are relatively fragile, maintenance-intensive, and costly to replace. Your group can't realistically make friends with some wandering mutt & turn it into a working dog in a few sessions. My experience is with training spaniels for field trials: it takes two years to raise a pup to a high working standard & it takes weekly training work to maintain that high working standard. I don't see military working dogs or any kind of detection dog being any different. This isn't something the RPG group can do in a game context. I think for game purposes I would expect to see more junkyard dogs/meat dogs than highly trained working dogs. But the possibility that there are highly trained working dogs out there adds some fun possibilities to a game.