The differentially private (DP) facility location problem seeks to determine a socially optimal placement for a public facility while ensuring that each participating agent's location remains private. To privatize its input data, a DP mechanism must inject noise into its output distribution, producing a placement that will have lower expected social welfare than the optimal spot for the facility. The privacy-induced welfare loss can be viewed as the "cost of privacy," illustrating a tradeoff between social welfare and privacy that has been the focus of prior work. Yet, the imposition of privacy also induces a third consideration that has not been similarly studied: fairness in how the "cost of privacy" is distributed across individuals. For instance, a mechanism may satisfy DP with minimal social welfare loss, yet still be undesirable if that loss falls entirely on one individual. In this paper, we quantify this new notion of unfairness and design mechanisms for facility location that attempt to simultaneously optimize across privacy, social welfare, and fairness. We first derive an impossibility result, showing that privacy and fairness cannot be simultaneously guaranteed over all possible datasets that could represent the locations of individuals in a population. We then consider a relaxation that still requires worst-case DP, but only seeks fairness and social welfare over smaller, more "realistic-looking" families of datasets. For this relaxation, we construct a DP mechanism and demonstrate that it is simultaneously optimal (or, for a harder family of datasets, near-optimal up to small factors) on fairness and social welfare. This suggests that while there is a tradeoff between privacy and each of social welfare and fairness, there is no additional tradeoff when we consider all three objectives simultaneously, provided that the population data is sufficiently natural.