Real-time information processing applications such as those enabling a more intelligent infrastructure are increasingly focused on analyzing privacy-sensitive data obtained from individuals. To produce accurate statistics about the habits of a population of users of a system, this data might need to be processed through model-based estimators. Moreover, models of population dynamics, originating for example from epidemiology or the social sciences, are often necessarily nonlinear. Motivated by these trends, this paper presents an approach to design nonlinear privacy-preserving model-based observers, relying on additive input or output noise to give differential privacy guarantees to the individuals providing the input data. For the case of output perturbation, contraction analysis allows us to design convergent observers as well as set the level of privacy-preserving noise appropriately. Two examples illustrate the approach: estimating the edge formation probabilities in a dynamic social network, and syndromic surveillance relying on an epidemiological model.