A variety of large-scale machine learning problems can be cast as instances of constrained submodular maximization. Existing approaches for distributed submodular maximization have a critical drawback: The capacity – the number of instances that can fit in memory of each machine – must grow with the data set size. In practice, while one can provision many machines, the capacity of each machine is limited by physical constraints. We propose a truly scalable approach for distributed submodular maximization when the capacity of each machine is fixed. The proposed framework applies to a broad class of algorithms and a variety of constraints. We provide theoretical guarantees on the approximation factor for any available capacity. We empirically evaluate the proposed algorithm on a variety of data sets and demonstrate that the algorithm achieves performance competitive with the centralized greedy solution.