We prove new complexity bounds for the primal-dual algorithm with random extrapolation and coordinate descent (PURE-CD), which has been shown to obtain promising practical performance for solving convex-concave min-max problems with bilinear coupling and dual separability. Such problems arise in many machine learning contexts, including empirical risk minimization, matrix games, and image processing. Our results either match or improve the best-known complexities of first-order algorithms for dense and sparse (strongly)-convex-(strongly)-concave problems with bilinear coupling.