In this work, synthesis and closed-loop operation of robust distributed model predictive control (MPC) for linear systems using distributed optimization is discussed. Previous work has shown that a nominal MPC controller for this setup can be synthesized and operated in a purely distributed manner. This paper extends this concept to linear systems subject to additive bounded disturbance. It is shown how well-established robust MPC approaches can be applied to distributed systems. The main focus of the paper is on a thorough discussion of computational issues arising from distributed synthesis and closed-loop operation of existing robust MPC controllers. In particular, techniques for distributed synthesis of structured robust positive invariant sets and distributed constraint tight- ening are proposed. The paper is concluded by a numerical example which illustrates the functionality and performance of the proposed techniques.