Having overseen the birth of three litters of puppies in the last two years I have been able to personally witness natural weaning and puppy socialization. Rather then give a clear cut age like 6 weeks or 9 weeks it is more like a range of time when puppies are ready to leave the nest. There's no set date - it depends on the puppy and the mother. It is essential that puppies have enough time to wean from their mother, learn to eat puppy kibble and begin learning how to act around other dogs. Good breeders will also give puppies their first set of vaccinations before releasing them, however this is not essential as long as the new owner is prepared to take the puppy to the veterinarian very shortly after taking the puppy home.
Newborn pups get all of their immunity and nutrition from their mother's milk. For the first 3 weeks of life they are pretty immobile, scooting only in the direction of their mother. If the mother and any other dogs on the premises have been kept on a regular vaccine schedule the puppies are indirectly receiving this immunity through the mother. As weaning occurs puppies become susceptible to contracting distemper, parvo virus and coronavirus, however the viruses must be present in the dogs they are around. So if the mother is healthy and she's the only dog other than the puppies there it is extremely unlikely that the puppies will become ill.
At about 3-4 weeks the puppies begin to get their tiny baby teeth, or milk teeth, and start to become more mobile. At this point the mother may start denying the puppies very frequent nursings, and as such it is a good time to introduce soft baby kibble to the puppies. At this age the puppies are very interested in each other and begin playing together. For the next 2-6 weeks they will grow, play and learn. Puppies that haven't been exposed to puppy kibble and go home before this is done risk going into shock from a blood sugar drop.
The absolute earliest a puppy should leave its mother and siblings is 6 weeks old, provided the puppy has had some time to wean from the mother and learn to eat puppy food. This is a perfect time to get the first vaccinations, whether the puppy has stopped weaning or not. Breeders who are keeping their puppies beyond 6 weeks should take the litter in to the vet and have them vaccinated. If the puppy has sufficiently weaned, shows an interest in puppy food and is observed playing with his siblings you can go ahead and take him to his new home.
Some puppies require longer to meet all these requirements. Puppies at 6, 7 and 8 weeks may have weaned but haven't had enough time to play with the other puppies and learn how to act around other dogs. Then again a mother with her first litter may not be willing to give up nursing until closer to 7 weeks and hasn't given the puppy an appropriate amount of time to become interested in puppy food. Or maybe they have done all these things but still appear skittish around humans and need some more socialization time. Whatever the case it is an individual thing among different puppies and even different litters from the same mom.
In the first litter the puppies were not ready to go to their new homes until about 8 weeks. The mother was a first time mommy , very good at her job. The second litter was a bit quicker - we took our puppy home at 6 weeks after the mother had weaned them several days prior. She had no problem socializing with other dogs and continues to be very active, social, happy and healthy. This third litter remains to be seen. At only 1 week old they are more similar to large rodents than puppies. However when they reach the 6-8 weeks range, depending on how they are doing, they will find new homes and loving human parents!