The availability of genomic data is often essential to progress in biomedical research, personalized medicine, drug development, etc. However, its extreme sensitivity makes it problematic, if not outright impossible, to publish or share it. As a result, several initiatives have been launched to experiment with synthetic genomic data, e.g., using generative models to learn the underlying distribution of the real data and generate artificial datasets that preserve its salient characteristics without exposing it. This paper provides the first evaluation of the utility and the privacy protection of six state-of-the-art models for generating synthetic genomic data. We assess the performance of the synthetic data on several common tasks, such as allele population statistics and linkage disequilibrium. We then measure privacy through the lens of membership inference attacks, i.e., inferring whether a record was part of the training data. Our experiments show that no single approach to generate synthetic genomic data yields both high utility and strong privacy across the board. Also, the size and nature of the training dataset matter. Moreover, while some combinations of datasets and models produce synthetic data with distributions close to the real data, there often are target data points that are vulnerable to membership inference. Looking forward, our techniques can be used by practitioners to assess the risks of deploying synthetic genomic data in the wild and serve as a benchmark for future work.