Mobile social apps provide sharing and networking opportunities based on a user's location, activity, and set of nearby users. A platform for these apps must meet a wide range of communication needs while ensuring users' control over their privacy. In this paper, we introduce EnCore, a mobile platform that builds on secure encounters between pairs of devices as a foundation for privacy-preserving communication. An encounter occurs whenever two devices are within Bluetooth radio range of each other, and generates a unique encounter ID and associated shared key. EnCore detects nearby users and resources, bootstraps named communication abstractions called events for groups of proximal users, and enables communication and sharing among event participants, while relying on existing network, storage and online social network services. At the same time, EnCore puts users in control of their privacy and the confidentiality of the information they share. Using an Android implementation of EnCore and an app for event-based communication and sharing, we evaluate EnCore's utility using a live testbed deployment with 35 users.