We propose the Flow Stochastic Segmentation Network (Flow-SSN), a generative model for probabilistic segmentation featuring discrete-time autoregressive and modern continuous-time flow parameterisations. We prove fundamental limitations of the low-rank parameterisation of previous methods and show that Flow-SSNs can estimate arbitrarily high-rank pixel-wise covariances without assuming the rank or storing the distributional parameters. Flow-SSNs are also more efficient to sample from than standard diffusion-based segmentation models, as most of the model capacity is allocated to learning the base distribution of the flow, which constitutes an expressive prior. We apply Flow-SSNs to challenging medical imaging benchmarks and achieve state-of-the-art results.