We quantify the cost of privacy in the federated setting by establishing matching lower and upper bounds, up to a logarithmic factor, on the minimax separation rate. This optimal rate benchmarks the difficulty of the testing problem, factoring in model characteristics such as the number of observations, noise level, and regularity of the signal class, along with the strictness of the (ε, δ)-DP requirement and the degree to which the data is distributed.

Our results demonstrate interesting and novel phase transition phenomena: where the cost of DP is minimal for testing in more centralized settings, it can significantly affect distributed scenarios. Furthermore, it is revealed that distributed one-shot protocols with access to shared randomness outperform those without access to shared randomness. We also construct a data-driven testing procedure that can adapt to an unknown regularity parameter over a large collection of function classes with minimal additional cost, while adhering to the same set of DP constraints.