We present a hydrodynamic theory describing pair diffusion in systems with periodic boundary conditions, thereby generalizing earlier work on self-diffusion [B. Dünweg and K. Kremer, J. Chem. Phys. 99, 6983-6997 (1993) and I.-C. Yeh and G. Hummer, J. Phys. Chem. B 108, 15873-15879 (2004)]. Its predictions are compared with Molecular Dynamics simulations for a liquid carbonate electrolyte and two ionic liquids, for which we characterize the correlated motion between distinct ions. Overall, we observe good agreement between theory and simulation data, highlighting that hydrodynamic interactions universally dictate ion correlations. However, when summing over all ion pairs in the system to obtain the cross-contributions to the total cationic or anionic conductivity, the hydrodynamic interactions between ions with like and unlike charges largely cancel. Consequently, significant conductivity contributions only arise from deviations from a hydrodynamic flow field of an ideal fluid, which is from the local electrolyte structure as well as the relaxation processes in the subdiffusive regime. In the case of ionic liquids, the momentum-conservation constraint additionally is vital, which we study by employing different ionic masses in the simulations. Our formalism will likely also be helpful to estimate finite-size effects of the conductivity or of Maxwell-Stefan diffusivities in simulations.