We consider a distributed function computation problem in which parties observing noisy versions of a remote source facilitate the computation of a function of their observations at a fusion center through public communication. The distributed function computation is subject to constraints, including not only reliability and storage but also privacy and secrecy. Specifically, 1) the remote source should remain private from an eavesdropper and the fusion center, measured in terms of the information leaked about the remote source; 2) the function computed should remain secret from the eavesdropper, measured in terms of the information leaked about the arguments of the function, to ensure secrecy regardless of the exact function used. We derive the exact rate regions for lossless and lossy single-function computation and illustrate the lossy single-function computation rate region for an information bottleneck example, in which the optimal auxiliary random variables are characterized for binary-input symmetric-output channels. We extend the approach to lossless and lossy asynchronous multiple-function computations with joint secrecy and privacy constraints, in which case inner and outer bounds for the rate regions differing only in the Markov chain conditions imposed are characterized.