Nonlinear aggregation is central to modern distributed systems, yet its privacy behavior is far less understood than that of linear aggregation. Unlike linear aggregation where mature mechanisms can often suppress information leakage, nonlinear operators impose inherent structural limits on what privacy guarantees are theoretically achievable when the aggregate must be computed exactly. This paper develops a unified information-theoretic framework to characterize privacy leakage in distributed nonlinear aggregation under a joint adversary that combines passive (honest-but-curious) corruption and eavesdropping over communication channels. We cover two broad classes of nonlinear aggregates: order-based operators (maximum/minimum and top-$K$) and robust aggregation (median/quantiles and trimmed mean). We first derive fundamental lower bounds on leakage that hold without sacrificing accuracy, thereby identifying the minimum unavoidable information revealed by the computation and the transcript. We then propose simple yet effective privacy-preserving distributed algorithms, and show that with appropriate randomized initialization and parameter choices, our proposed approaches can attach the derived optimal bounds for the considered operators. Extensive experiments validate the tightness of the bounds and demonstrate that network topology and key algorithmic parameters (including the stepsize) govern the observed leakage in line with the theoretical analysis, yielding actionable guidelines for privacy-preserving nonlinear aggregation.