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Image segmentation is important in medical imaging, providing valuable, quantitative information for clinical decision-making in diagnosis, therapy, and intervention. The state-of-the-art in automated segmentation remains supervised learning, employing discriminative models such as U-Net. However, training these models requires access to large amounts of manually labelled data which is often difficult to obtain in real medical applications. In such settings, semi-supervised learning (SSL) attempts to leverage the abundance of unlabelled data to obtain more robust and reliable models. Recently, generative models have been proposed for semantic segmentation, as they make an attractive choice for SSL. Their ability to capture the joint distribution over input images and output label maps provides a natural way to incorporate information from unlabelled images. This paper analyses whether deep generative models such as the SemanticGAN are truly viable alternatives to tackle challenging medical image segmentation problems. To that end, we thoroughly evaluate the segmentation performance, robustness, and potential subgroup disparities of discriminative and generative segmentation methods when applied to large-scale, publicly available chest X-ray datasets.