Abstract We study the effects of property rights over the use of data on market outcomes. To do so, we consider a model in which a monopolistic firm offers a service to a set of heterogeneous users. The use of the service generates valuable data, but data monetization entails a privacy cost for users. A trade-off emerges between under-processing and over-monetization of data. We show that both the firm and users prefer the users (the firm) to own the rights for low (high) values of data. We further discuss the robustness of our results when allowing more possible contracts for the data owner and show that the main trade-off is robust to these extensions.