Personal Data Management System (PDMS) solutions are flourishing, boosted by smart disclosure initiatives and new regulations. PDMSs allow users to easily store and manage data directly generated by their devices or resulting from their (digital) interactions. Users can then leverage the power of their PDMS to benefit from their personal data, for their own good and in the interest of the community. The PDMS paradigm thus brings exciting perspectives by unlocking novel usages, but also raises security issues. An effective approach, considered in several recent works, is to let the user data distributed on personal platforms, secured locally using hardware and/or software security mechanisms. This paper goes beyond the local security issues and addresses the important question of securely querying this massively distributed personal data. To this end, we propose DISPERS, a fully-distributed PDMS peerto-peer architecture. DISPERS allows users to securely and efficiently share and query their personal data, even in the presence of malicious nodes. We consider three increasingly powerful threat models and derive, for each, a security requirement that must be fulfilled to reach a lower-bound in terms of sensitive data leakage: (i) hidden communications, (ii) random dispersion of data, and (iii) collaborative proofs. These requirements are incremental and respectively resist spied, leaking or corrupted nodes. We show that the expected security level can be guaranteed with near certainty and validate experimentally the efficiency of the proposed protocols, allowing for adjustable trade-off between the security level and its cost.