We discuss primal-dual methods for MFG and MFC problems with mixed, local-nonlocal, couplings. Introducing dual variables for couplings of each type, we formulate the MFG system as a monotone inclusion and solve it via an extension of the primal-dual hybrid gradient (PDHG) algorithm. Building on our earlier work, we choose the dual variables for nonlocal terms in a Fourier space. Thus, we completely split all types of interactions in the system. Another appealing feature of our method is that it also handles non-potential MFGs with monotone couplings. To the best of our knowledge, this is the first instance of applying primal-dual optimization methods to non-potential MFG systems. This work is joint with Siting Liu.