What kind of worms? Not to put you off eating, or anything, but.... do they look like spaghetti or grains of rice?
If they look like spaghetti, then they are probably round worms, and yes, they can pose a threat to humans, especially small children. If a puppy is expelling lots of round worms, then they probably have an extremely heavy burden, which can take a toll on their health, as well.
If they look like grains of rice (especially when dry) then they are probably tape worms, usually caused by ingesting infected fleas.
Consulting with your vet about getting the proper medication for treatment would be your best course of action. He could prescribe two different medications, or he could prescribe a broad spectrum dewormer. In either case, you need to be super stringent about picking up poop, and making sure that you follow the schedule for repeating the medication, as it usually takes two to four rounds of dewormer before they are eliminated. If he has tapeworms, you will also probably need to use a flea treatment