This paper investigates the dependence of existing state-of-the-art person re-identification models on the presence and visibility of human faces. We apply a face detection and blurring algorithm to create anonymized versions of several popular person re-identification datasets including Market1501, DukeMTMC-reID, CUHK03, Viper, and Airport. Using a cross-section of existing state-of-the-art models that range in accuracy and computational efficiency, we evaluate the effect of this anonymization on re-identification performance using standard metrics. Perhaps surprisingly, the effect on mAP is very small, and accuracy is recovered by simply training on the anonymized versions of the data rather than the original data. These findings are consistent across multiple models and datasets. These results indicate that datasets can be safely anonymized by blurring faces without significantly impacting the performance of person re-identification systems, and may allow for the release of new richer re-identification datasets where previously there were privacy or data protection concerns.