This research takes an information theoretic approach to develop principles that govern secure or private function computation by multiple terminals that host user data. The goal of the terminals is to compute locally and reliably, a given function of all the possibly correlated user data, using an interactive communication protocol. The protocol is required to satisfy separate security and privacy conditions. The former stipulates for each terminal that a coalition of the remaining terminals should glean no more information about the data at the terminal from their own data and the communication -- than can be obtained from the function value. The latter protects each individual user's data at a terminal from a similar coalition. A common framework is developed for analyzing the distinct concepts of security and privacy, and new information theoretic formulations and approaches are proposed with the objective of understanding basic underlying principles. Potential applications arise, for instance, in: hospital databases that store clinical drug trial results or university databases with student performance records; private information retrieval from user data stored in private clouds; and security and privacy certifications for the identities/locations of communities and individuals participating in crowd-sourced traffic and navigation services.