Aiming to combine density functional theory (DFT) and wavefunction theory, we study a mapping from the many-body interacting system to an effectively-interacting Kohn-Sham system instead of a non-interacting Kohn-Sham system. Because a ground state of effectively-interacting systems requires having a solution for the correlated many-body wavefunctions, this provides a natural framework to many-body wavefunction theories such as the configuration interaction and the coupled cluster method in the formal theoretical framework of DFT. Employing simple one-dimensional two-electron systems -- namely, the one-dimensional helium atom, hydrogen molecule and heteronuclear diatomic molecule -- we investigate properties of many-body wavefunctions and exact exchange-correlation potentials of effectively-interacting Kohn-Sham systems. As a result, we find that the asymptotic behavior of the exact exchange-correlation potential can be controlled by optimizing that of the effective interaction. Furthermore, the typical features of the exact non-interacting Kohn-Sham system, namely a spiky feature and a step feature in the exchange-correlation potential for the molecular dissociation limit can be suppressed by a proper choice of the effective interaction. These findings open a possibility to construct numerically robust and efficient exchange-correlation potentials and functionals based on the effectively-interacting Kohn-Sham scheme.