Privacy concerns around sharing personally identifiable information are a major practical barrier to data sharing in medical research. However, in many cases, researchers have no interest in a particular individual's information but rather aim to derive insights at the level of cohorts. Here, we utilize Generative Adversarial Networks (GANs) to create derived medical imaging datasets consisting entirely of synthetic patient data. The synthetic images ideally have, in aggregate, similar statistical properties to those of a source dataset but do not contain sensitive personal information. We assess the quality of synthetic data generated by two GAN models for chest radiographs with 14 different radiology findings and brain computed tomography (CT) scans with six types of intracranial hemorrhages. We measure the synthetic image quality by the performance difference of predictive models trained on either the synthetic or the real dataset. We find that synthetic data performance disproportionately benefits from a reduced number of unique label combinations and determine at what number of samples per class overfitting effects start to dominate GAN training. Our open-source benchmark findings also indicate that synthetic data generation can benefit from higher levels of spatial resolution. We additionally conducted a reader study in which trained radiologists do not perform better than random on discriminating between synthetic and real medical images for both data modalities to a statistically significant extent. Our study offers valuable guidelines and outlines practical conditions under which insights derived from synthetic medical images are similar to those that would have been derived from real imaging data. Our results indicate that synthetic data sharing may be an attractive and privacy-preserving alternative to sharing real patient-level data in the right settings.