Real-time signal processing applications are increasingly focused on analyzing privacy-sensitive data obtained from individuals, and this data might need to be processed through model-based estimators to produce accurate statistics. Moreover, the models used in population dynamics studies, e.g., in epidemiology or sociology, are often necessarily nonlinear. This paper presents a design approach for nonlinear privacy-preserving model-based observers, relying on contraction analysis to give differential privacy guarantees to the individuals providing the input data. The approach is illustrated in two applications: estimation of edge formation probabilities in a dynamic social network, and syndromic surveillance relying on an epidemiological model.