Discrete Duality Finite Volume (DDFV) methods are very well suited to discretize anisotropic diffusion problems, even on meshes with low mesh quality. Their performance stems from an accurate reconstruction of the gradients between mesh cell boundaries, which comes however at the cost of using both a primal (cell centered) and a dual (vertex centered) mesh, and thus leads to larger system sizes. To solve these systems, we propose to use non-overlapping optimized Schwarz methods with Robin transmission conditions, which can also well take into account anisotropic diffusion across subdomain interfaces. We study these methods here directly at the discrete level, and prove convergence using energy estimates for general decompositions including cross points and fully anisotropic diffusion. Our analysis reveals that primal and dual meshes might be coupled using different optimized Robin parameters in the optimized Schwarz methods. We present both the separate and coupled optimization of Robin transmission conditions and derive parameters which lead to the fastest possibe convergence in each case. We illustrate our analysis with numerical experiments for the model problem covered by our analysis, and also in situations that go beyond our analysis, with an application to anisotropic image reconstruction.