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We present a scalable distributed memory library for generating and computations involving structured dense matrices, such as those produced by boundary integral equation formulations. Such matrices are dense, but have special structure that can be exploited to obtain efficient storage and matrix-vector product evaluations and consequently the fast solution of linear systems. At the core of the methods we use is the observation that off-diagonal matrix blocks of such matrices have a low numerical rank, and that this property can be exploited in a multi-level fashion. In this work we focus on the Hierarchically Semi-Separable (HSS) representation. We present algorithms for building and using HSS representations that are parallelized using MPI and CUDA to leverage state-of-the-art heterogeneous clusters. The efficiency of our methods and implementation is demonstrated on large dense matrices obtained from a boundary integral equation formulation of the Laplace equation with Dirichlet boundary conditions. We demonstrate excellent (linear) scalability on up to 128 GPUs on 128 nodes. Our codes will lay the foundation for fast direct solvers for elliptic problems.