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We analyze to what extent final users can infer information about the levelof protection of their data when the data obfuscation mechanism is a prioriunknown to them (the so-called ''black-box'' scenario). In particular, we delveinto the investigation of two notions of local differential privacy (LDP),namely {\epsilon}-LDP and R\'enyi LDP. On one hand, we prove that, without anyassumption on the underlying distributions, it is not possible to have analgorithm able to infer the level of data protection with provable guarantees;this result also holds for the central versions of the two notions of DPconsidered. On the other hand, we demonstrate that, under reasonableassumptions (namely, Lipschitzness of the involved densities on a closedinterval), such guarantees exist and can be achieved by a simplehistogram-based estimator. We validate our results experimentally and we notethat, on a particularly well-behaved distribution (namely, the Laplace noise),our method gives even better results than expected, in the sense that inpractice the number of samples needed to achieve the desired confidence issmaller than the theoretical bound, and the estimation of {\epsilon} is moreprecise than predicted.

On the (Im)Possibility of Estimating Various Notions of Differential Privacy