Any moisture in paper causes it to become soggy and pulpy. We used it for cat litter for a while. It retains all odors and moisture. It tracks all over the place when in contact with fur and paws. And, I agree with everyone else, the puppy will likely eat it. Our former dog, a mini dachshund, found it a great coating for Kitty Roca. Kinda turned it into Breaded Kitty Roll.
If you are looking to save money, go to a Dollar Store and get a couple of their baby blankets. They wash and dry quickly and will allow you to have economical bedding that is fresh and safe for your puppy.
If you're looking for a way to use old paper you could use it for paper training but the same odor and moisture problems will occur and the any ink will come off on the puppy's paws and onto floors and carpets. Pee pads cost a bit but will pay off in training and reduced damage to your home.