Peer-to-peer data integration - a.k.a. Peer Data Management Systems (PDMSs) - promises to extend the classical data integration approach to the Internet scale. Unfortunately, some challenges remain before realizing this promise. One of the biggest challenges is preserving the privacy of the exchanged data while passing through several intermediate peers. Another challenge is protecting the mappings used for data translation. Protecting the privacy without being unfair to any of the peers is yet a third challenge. This paper presents a novel query answering protocol in PDMSs to address these challenges. The protocol employs a technique based on noise selection and insertion to protect the query results, and a commutative encryption-based technique to protect the mappings and ensure fairness among peers. An extensive security analysis of the protocol shows that it is resilient to several possible types of attacks. We implemented the protocol within an established PDMS: the Hyperion system. We conducted an experimental study using real data from the healthcare domain. The results show that our protocol manages to achieve its privacy and fairness goals, while maintaining query processing time at the interactive level.