As stated in the Buy section, avoid purchasing sunglasses, umbrellas, and other trinkets from illegal sellers. Definitely steer clear of counterfeit luxury goods. Upon arriving to Pisa, it is not uncommon for tourists to be swarmed by vendors. Some vendors can be pushy or even downright aggressive. Even saying "no thank you" can bring unwanted attention. It's much better to ignore them entirely. Don't worry about being rude.
Even though cheap prices are tempting, especially when traveling on a budget, please remember that recently the Italian police have fined tourists for buying from illegal sellers. These fines can be very steep up to €1000. In front of the Tower, there are several legal vendors who have permits to sell items to tourists or anyone else. Be responsible and buy from them. Legal vendors have stalls lined along the main road whereas illegal vendors keep merchandise in a sack, cardboard boxes, or in the case of fake purses on their arms.