We propose an analytical framework to tightly characterize the scaling laws for the additional bandwidth that servers must supply to guarantee perfect service in peer-assisted Video-on-Demand systems, taking into account essential aspects such as peer churn, bandwidth heterogeneity, and Zipf-like video popularity. Our results reveal that the catalog size and the content popularity distribution have a huge effect on the system performance. We show that users' cooperation can effectively reduce the servers' burden for a wide range of system parameters, confirming to be an attractive solution to limit the costs incurred by content providers as the system scales to large populations of users.