A basic model for key agreement with a remote (or hidden) source is extended to a multi-user model with joint secrecy and privacy constraints over all entities that do not trust each other after key agreement. Multiple entities using different measurements of the same source through broadcast channels (BCs) to agree on mutually-independent local secret keys are considered. Our model is the proper multi-user extension of the basic model since the encoder and decoder pairs are not assumed to trust other pairs after key agreement, unlike assumed in the literature. Strong secrecy constraints imposed on all secret keys jointly, which is more stringent than separate secrecy leakage constraints for each secret key considered in the literature, are satisfied. Inner bounds for maximum key rate, and minimum privacy-leakage and database-storage rates are proposed for any finite number of entities. Inner and outer bounds for degraded and less-noisy BCs are given to illustrate cases with strong privacy. A multi-enrollment model that is used for common physical unclonable functions is also considered to establish inner and outer bounds for key-leakage-storage regions that differ only in the Markov chains imposed. For this special case, the encoder and decoder measurement channels have the same channel transition matrix and secrecy leakage is measured for each secret key separately. We illustrate cases for which it is useful to have multiple enrollments as compared to a single enrollment and vice versa.