Should a puppy with its first vaccinations meet a dog that’s not fully vaccinated?

I'm sorry but the answer is no, you shouldn't have them meet yet. In fact, non-vaccinated young puppies should be kept away from other dogs as much as possible until they have received all of their vaccinations.

The risk is that the other dog, who is an adult and has a stronger immune system, could be carrying some illness without showing symptoms. Your young puppy would have few protections and could possibly get quite sick if this was the case.

Now, the odds of this happening are fairly low. However, it is probably not worth the risk. If you had a 1/1000 chance of your puppy dying because you wanted to show him off, would you take that chance?

Summary: in contrast to the other answers, I would not worry too much about the infection risk - although explicitly finding out that the other dog has not shown any signs of disease lately is a sensible precaution.

Update: turns out the puppy is much younger than I thought - it will move to Tom just a few days before Christmas, presumably at the usual 9 - 10 weeks age. Regardless of vaccination, for a puppy of 2 1/2 months ca. 4 days after moving into the new home is IMHO too early to expose it again to a change of places and lots of new people - I'm not thinking infection risk here but stress and anxiety. At that point, the puppy needs calm and reliable surroundings, and OP to look after it.
(end of update)

---

Some initial thoughts:

• How old is the not-boosted dog now and when did it get is 1st + 2nd shot (i.e. if you speak of the 8 and 12 week vaccinations was the dog actually 2/3 months old or did it possibly get the first 2 shots later)? If the other dog is not very old yet and it got its shots when somewhat older (e.g. at 3 + 4 months age or 4 + 6 months) the risk that this dog carries one of those infectious diseases will still be much lower than with a totally unvaccinated dog.

• Your question sounds as if you already have your puppy. If so,

  • what's the reason it did not yet have its 1st vaccination? Over here, the first vaccination is usually done ca. 1 week before the puppy goes to their new owner.
  • The 1st vaccination is usually followed up rather quickly by the second (4 weeks later) is that when vaccinating early 8 (= before they leave their litter) mostly because there's some risk that the 1st vaccination isn't very effective as the puppy still has maternal antibodies (via milk). If the puppy is with you already, that should not be a concern in your case.
  • Other than that, the 1st vaccination does give immunity - it just won't last very long. However, we're talking about 2-4 weeks after that immunization and at that time your puppy should be fine with other dogs. Also, a level of immunity that is not sufficient to totally prevent infection often still helps in keeping the infection less severe.

• What's the alternative? If you stay strictly at home with the puppy instead, that's clearly lower risk of meeting infections. But already if you treat yourself and the puppy to some short walks in the forest, that may come with a much higher risk compared to having the dog play with a single individual (whos vaccination status may be much better than you fear).
  If the dog would have to go to puppy daycare instead the risk there would probably be much higher than meeting one other dog.

• As you know of the other dog's vaccination "habits", you probably know a bit more about that dog. Personally, I'd not be worried about a dog that occasionally meets other dogs on the walk - but would be more careful with a dog that, say, regularly goes to dog pension. And, maybe you could explain that your puppy will only have a recent 1st vaccination and ask the owners of the other dog to tell you whether there were/are any signs of illness recently - knowing that the dog is and was healthy cannot totally exclude disease, but the risk is obviously much lower.

I assume you wouldn't think of bringing your puppy into contact with the other dog unless you consider the other dog's owners in general trustworthy and reliable (e.g., the other dog is not a recently illegally imported feral dog from some other country).

Let's go through the list of things dogs are vaccinated against (here in Europe). I'll start with the scary stuff.

• Rabies. Definitively something I want my dog as well as other dogs to be vaccinated against (and btw we even had a wildlife vaccination campaign over 25 years in order to get rid of rabies [not the bat version, though]). However, in the last 10 years, AFAIK we had 3 cases of rabid dogs that were illegaly imported (vaccination is required on import) and 1 case of human rabies (dog bite in North Africa). Also, while a recent rabies vaccination is officially required for dogs crossing borders around here, the boosting recommendation is every 2 - 3 years.
  Frankly, if the other dog has rabies, your puppy is your least concern. If you live in a country where rabies still occurs, the other dog in the family is also most probably of far less concern than wildlife/feral dogs your puppy meets when leaving the house.

• Canine distemper. That's measles for canides and it is dangerous (4 out of 5 infected puppies with clinical symptom die, surviving puppies may have lifelong disabilities and may have to be euthanazed due to long-term consequences). I read up on distemper when I got my dog/puppy 2 years ago because we had canine distemper in the wildlife - and fox, racoon & Co. do come right up to our house. I learned that if there's a really high danger of distemper for the puppy, a shot with human measles vaccine does help: that vaccine is not disactivated by dog maternal antibodies but it causes sufficient immunity to prevent severe forms. Once the basic vaccination is done, the recommendation here is to have boosting shot every 3 years - yearly is not necessary. In my region I'd consider wildlife to pose a much higher risk.

• Parvovirosis: another nasty one. However, also caused by a virus keeps alive under environmental conditions for years and it can be spread via human (transporting it) from an infected dog to another dog. Not only canides but also e.g. cats and IIRC martens can get/have/spread it. Again, everyday occasions are likely to have a higher overall risk than the single well-known family dog, boosting recommendation every 3 years.

• Leptospirosis: again tons of other animals ranging from mice over rabbits, hedgehogs, deer, cow, sheep, pigs, raccoons etc. all the way to humans are the relevant reservoir. Moist urine is infectious, and so is contaminated soil and water. This is the only of the core dog vaccinations where annual boosting is recommended: even after infection, immunity doesn't last very long and there are various serotypes around (meaning also that properly vaccinated dogs can have and spread leptospirosis).
  Sounds like things you/your puppy meets everyday?

• Infectious canine hepatitis is fortunately rare nowadays around here. Another one where the virus may survive for months, boosting recommended every 3 years. (Have to run now - will update when I have time)

• Kennel cough is very infectious. Its main cause, Bordetella bronchiseptica is closely related to the bacteria causing whooping cough in humans. Another one that lasts long under environmental conditions. Not only canides but also cats, pigs, rabbits can get it. Vaccination against kennel cough is recommended only for dogs who have an elevated risk of exposure (dog pension), but it is not considered necessary for every dog. Again, if you puppy got kennel cough, chances are it wasn't the other family dog but rather picked up on the walk.

---

Here are the recommendations of the German veterinary vaccination commission (in German, though) which I used to look up boosting recommendations.