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In this work, we construct a structured framework for the efficient random sampling and recovery of bandlimited graph signals that lie on product graphs. Product graphs are a model to construct large complex graphs from smaller simpler building blocks we call graph atoms, and are a convenient tool to model rich classes of multi-modal graph-structured data. Our randomized sampling framework prescribes an optimal sampling distribution over the nodes of the product graph constructed by only processing these smaller graph atoms. As a result, the framework achieves significant savings in computational complexity with respect to previous works that do not exploit the inherent structure of product graphs.