Local differential privacy is a model for privacy in which an untrusted statistician collects data from individuals who mask their data before revealing it. While randomized response has shown to be a good strategy when the statistician's goal is to estimate a parameter of the population, we consider instead the problem of locally private data publishing, in which the data collector must publish a version of the data it has collected. We model utility by a distortion measure and consider privacy mechanisms that act via a memoryless channnel operating on the data. If we consider a the source distribution to be unknown but in a class of distributions, we arrive at a robust-rate distortion model for the privacy-distortion tradeoff. We show that under Hamming distortions, the differential privacy risk is lower bounded for all nontrivial distortions, and that the lower bound grows logarithmically in the alphabet size.