We consider the following stochastic model for a mobile service scenario. Consider a stationary Poisson process in Rd, with its points radially ordered with respect to the origin (the anchor); if d = 2, the points may correspond to locations of e.g. restaurants. A user, with a location different from the origin, asks for the location of the first Poisson point and keeps asking for the location of the next Poisson point until the first time that he can be completely certain that he knows which Poisson point is his nearest neighbour. This waiting time is the communication cost, while the inferred privacy region is a random set obtained by an adversary who only knows the anchor and the points received from the server, where the adversary ‘does the best' to infer the possible locations of the user. Probabilistic results related to the communication cost and the inferred privacy region are established for any dimension d ≥ 1. Furthermore, special results when d = 1 and particularly when d = 2 are derived.