arXiv:2610.41034v1 Announce Type: new Abstract: Consider distributed databases that contain both sensitive (private) and non-sensitive (public) attributes. A user seeks to compute a function of public attributes and side information by interacting with the databases and downloading their responses. In this paper, we study the trade-offs among download cost, information leakage about the private attributes, and distortion between the computed function and its reconstruction, referred to as the capacity region. Our main contributions are as follows. For a single database, we provide a complete characterization of the capacity region for general discrete sources and characterize the capacity region of two different setups for Gaussian sources when the computed function is defined as the sum of the public attribute and side information. For distributed databases, we establish inner and outer bounds on the capacity region of discrete sources and we characterize the capacity region when the databases and the side information are mutually independent. As a special instance, we study lossless function computation, and results analogous to the lossy case are obtained.