Over the last five years, classical and optimized Schwarz methods have been developed for anisotropic elliptic problems discretized with Discrete Duality Finite Volume (DDFV) schemes. Like for Discontinuous Galerkin methods (DG), it is not a priori clear how to appropriately discretize transmission conditions with DDFV, and numerical experiments have shown that very different scalings both for the optimized parameters and the contraction rates of the Schwarz algorithms can be obtained, depending on the discretization. We explain in this article how the DDFV discretization can influence the performance of the Schwarz algorithms, and also propose and study a new DDFV discretization technique for the transmission conditions which leads to the expected convergence rate of the Schwarz algorithms obtained from an analysis at the continuous level.