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Distributed optimization methods for Optimal Power Flow (OPF) problems are of importance in reducing coordination complexity and ensuring economic grid operation. Renewable feed-ins and demands are intrinsically uncertain and often follow non-Gaussian distributions. The present paper combines uncertainty propagation via Polynomial Chaos Expansion (PCE) with the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method to solve stochastic OPF problems with non-Gaussian uncertainties in a distributed setting. Moreover, using ALADIN and PCE we obtain fast convergence while avoiding computationally expensive sampling. A numerical example illustrates the performance of the proposed approach.