Generative Adversarial Networks (GANs) are among the most popular approaches to generate synthetic data, especially images, for data sharing purposes. Given the vital importance of preserving the privacy of the individual data points in the original data, GANs are trained utilizing frameworks with robust privacy guarantees such as Differential Privacy (DP). However, these approaches remain widely unstudied beyond single performance metrics when presented with imbalanced datasets. To this end, we systematically compare GANs trained with the two best-known DP frameworks for deep learning, DP-SGD, and PATE, in different data imbalance settings from two perspectives -- the size of the classes in the generated synthetic data and their classification performance. Our analyses show that applying PATE, similarly to DP-SGD, has a disparate effect on the under/over-represented classes but in a much milder magnitude making it more robust. Interestingly, our experiments consistently show that for PATE, unlike DP-SGD, the privacy-utility trade-off is not monotonically decreasing but is much smoother and inverted U-shaped, meaning that adding a small degree of privacy actually helps generalization. However, we have also identified some settings (e.g., large imbalance) where PATE-GAN completely fails to learn some subparts of the training data.