How old is your puppy?
What age was your puppy when the first two shots were given?
I just wanted to chime in (I agree with what Dr D said) but thought perhaps I could give you a little more information to help you.
Let's start at the beginning...The mother passes on antibodies to her pups in the colostrum (the first milk produced). If the mother is well vaccinated and healthy, and if the pups are in good shape and get their share of the colostrum, they generally have received enough of the maternal antibodies to protect them for the first few months of their lives. During this time any vaccines we administer probably don't affect the pup's immune system much. We do know that sometime between 7 and 15 weeks the level of maternal immunity starts to decrease - the problem is that you cannot tell from looking at the pup where they stand.
On the pups side, their immune systems start maturing as they grow, and somewhere during that 7-15 week period their immune systems are actually able to respond to vaccine and develop the different antibodies needed to protect against these diseases. One type of cell that develops is a "memory cell" and these help the body to remember the diseases. The boosters also help to increase the level and memory of these cells, so we get a much quicker response. Again, this process is also affected by the pup's health: stress from overcrowding, malnutrition, moving to new homes, and parasitism can affect the pup's immune system and their ability to respond to the vaccines. And again, we can't tell just from looking at the pup how well they will respond. Thus, the boostering process.
Another factor to also consider is the vaccine itself. Quality vaccine, proper storage & handling, and proper administration are all factors that can affect the level of immune response. Good vaccine companies will also make changes in their vaccine to reflect the changes in the viruses/bacteria in the real world.
So, finally to answer your question, I wouldn't say that the initial vaccinations are worthless, we just do not know how effective that they are. Personally, I am not a big fan of titres - not only are they pretty expensive, they information can be very limited (like looking at a poloroid picture and trying to explain the plot of the whole movie). It generally is easier to simply provide extra vaccinations both for efficacy and peace of mind of all involved. As for being around other animals, as long as you know that they are healthy and well-vaccinated, go for it! This is the time when your pup needs as much socialization time as possible (and what fun to go to a friends house for coffee and a play day!)
I hope that this helps give you a little more information and helps with your decisions. Have fun with your little boy!