It actually depends on WHEN the other pup got his shots, not how many times he's had them. Nursing puppies get antibodies from their mother's milk, which serves as disease protection when they are very young. This immunity fades after the pups are weaned. A puppy can retain maternal antibodies for anywhere from 1 to 4 weeks after being weaned, but there's no real way to know EXACTLY when that protection is gone and the puppy needs to be vaccinated. If vaccinated too early, the pup's maternal antibodies destroy the vaccine; it doesn't work at all. If vaccinated too late, you could be leaving the pup exposed to diseases for weeks with absolutely no protection. This is the reason for the vaccination series that puppies get. Depending on when the pups were weaned, it's very likely that their first round of vaccinations don't work at all. Come time for the second round, some pups will have lost their maternal antibodies and they will benefit from the vaccine at that time; others who still have immunity from mom will not be affected, same as the first set of shots. By the time the last shots are due, it can be assumed that the vast majority of pups have been off mom's milk for
at least a month, and the vaccines will work like they are supposed to for all puppies. Some vets even do a 4th round of shots for pups.
So... if your friend's pup had his first round of shots very recently, they probably worked and the dog is protected against those diseases. BUT if the first shots were given back when the pup was tiny, they might not have worked.
Just a little "Puppy Vax 101"
IMO, as long as the other pup hasn't been out and about in public places, being exposed to many other dogs or dog areas, he should be fine.