You need to wait 2 weeks after the final vaccination to allow your puppy to mix with dogs that haven't been fully vaccinated. For socialisation it's good to find dogs who have been fully vaccinated or other puppies that are similar age and are also starting their vaccinations.
For yearly boosters, it would depend when the dog had its last yearly booster. Some vaccinations provide immunity for a lifetime, or many years. Kennel cough is the main concern for dogs that are not having booster shots. It's impossible to know if the dog is immune or not without a titre test. Also the puppy does have some immunity with the commencement of their vaccinations.
You could always ask them to get the dog's booster vaccinations. Or one for Kennel cough.
Unless or course the other dog doesn't go into public places or have other dogs visiting. In which case it's unlikely the dog would have any disease.
To be on the absolute safe side, it would be better to keep them separated.
Some reading:
https://www.innersouthvets.com.au/dog-vaccinations-annual-or-three-yearly/
You might have read about the changes to canine vaccinations in the media over the past few years. Research has emerged that the core vaccinations against parvo, distemper and hepatitis viruses (C3) may not need to be given annually as previously thought. Once the dogs are adults a booster vaccination every three years for parvo, distemper and hepatitis viruses may be sufficient to protect your pet. (This is NOT the case for Kennel Cough vaccination however). This is quite similar to the changes that we see to human vaccination schedules from time to time. As science evolves, so too do the schedules recommended to protect our beloved canine family members!
When can I take my puppy outside?
Vets recommend waiting until one to two weeks after your puppy’s last vaccination booster – usually at around 14–16 weeks of age – before introducing them to the wonders of local parks, beaches and walking trails.
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Start by introducing your puppy to known ‘safe’ dogs – dogs that you know have a good temperament and that have been fully vaccinated. It’s best to do this in a safe environment such as your backyard. You can also try socialising your puppy by introducing them to lots of different people and by giving puppy school a whirl.
Puppy and dog vaccination schedules and vaccination costs
Over the last decade there has been increasing debate about the over-vaccination of pets. Depending on the vaccine used and the individual animal, immunity can last significantly longer than 12 months. Some pet owners worry about vaccinating their pet whilst its immunisation is still sufficient.
If you are concerned about this, and would like to test if your dog’s immunisation is still sufficient, an antibody titre test may be an option to consider. In this process, a small amount of blood is taken from your pet and a laboratory test is used to determine the level of antibodies in your dog’s body.
Still Vaccinating Your Pet Every Year?
"We know that for [canine] distemper and parvo, for example, the immunity lasts a minimum of five years, probably seven to nine years, and for some individuals for a lifetime,” says veterinarian Jean Dodds, founder of Hemopet, the first nonprofit national blood bank program for animals, located in Santa Monica, Calif.
“For cats, so far we have challenge data out nine years showing that immunity is still protective," says Dodds. And with rabies vaccine, new data indicate the immunity lasts for at least seven years, she says.