Should rights be publicly or privately enforced in the case of digital piracy? The emergence of large-scale anti-piracy laws and the existence of illegal non-monitored channels raise important issues for the design of anti-piracy policies. We study the impact of these demand-side policies in two enforcement settings (namely, public and private enforcement settings) with an outside adoption option for users of an illegal non-monitored channel. Our results show that public enforcement generates higher monitoring and lower price levels, and also higher legal welfare than private enforcement. However, we identify potential conflicts of interest between the legal seller and the social planner when the efficiency of the illegal non-monitored channel is low. Introducing supply-side policies, i.e. policies targeted to suppliers of illegal content, we find that they may have unexpected impacts and can damage legal welfare. We also identify situations in which the two policies are substitutes or complements.