Urban transportation is being transformed by mobility-on-demand (MoD) systems. One of the goals of MoD systems is to provide personalized transportation services to passengers. This process is facilitated by a centralized operator that coordinates the assignment of vehicles to individual passengers, based on location data. However, current approaches assume that accurate positioning information for passengers and vehicles is readily available. This assumption raises privacy concerns. In this work, we address this issue by proposing a method that protects passengers' drop-off locations (i.e., their travel destinations). Formally, we solve a batch assignment problem that routes vehicles at obfuscated origin locations to passenger locations (since origin locations correspond to previous drop-off locations), such that the mean waiting time is minimized. Our main contributions are two-fold. First, we formalize the notion of privacy for continuous vehicle-to-passenger assignment in MoD systems, and integrate a privacy mechanism that provides formal guarantees. Second, we present a polynomial-time iterative version of the Hungarian algorithm to allocate a redundant number of vehicles to a single passenger. This algorithm builds on the insight that even during peak rush hour there are unoccupied (redundant) traveling vehicles. This strategy allows us to reduce the performance deterioration induced by the privacy mechanism. In particular, it enables the exploration of the trade-off between privacy levels, waiting time, and deployed fleet size. We evaluate our methods on a real, large-scale data set consisting of over 11 million taxi rides (specifying vehicle availability and passenger requests), recorded over a month's duration, in the area of Manhattan, New York. Based on current traffic statistics, our evaluations indicate that privacy can be achieved without incurring a significant loss of performance, and that this loss can be further controlled by varying operator or user preferences.