
There are no checks and no qualifications, (that is why there are so many pedigree dogs of poor breeding that bear only passing resemblance to their breed and some may not even be purebred)the only thing that needs to happen is for both parents to be the same breed and KC registered and a form to be signed by the stud owner confirming the mating.
The bitch owner (breeder) fills out the form with the chosen names pays the fee per pup, and receives the registration certificates as the breeder and owner of each pup.
When you sell a puppy you sign the transfer part of the form, and the new owner is supposed to send it off to KC with another fee to get it into their name. Many pet owners don't bother as it has nothing to do with legal ownership.
With the sheer numbers bred each year the KC would need an army of people checking litters (though personally I think it would be a good thing).
I am going to try and find out how many puppies of all breeds are registered each year.