This paper considers networks of agents that seek to cooperatively solve a general class of nonsmooth convex optimization problems with an inherent distributed structure. We characterize the asymptotic convergence properties of distributed continuous-time coordination algorithms whose design relies on the saddle-point dynamics associated with an augmented Lagrangian. The main technical novelty is the identification of a nonsmooth Lyapunov function which, under mild convexity and regularity assumptions on the optimization problem data, allows us to further characterize the exponential convergence rates of the proposed algorithms for optimization subject to either equality or inequality constraints.