We have measured total absolute cross sections for the mutual neutralization (MN) of O- with O+ and N+. A fine resolution (of about 50 meV) in the kinetic energy spectra of the product neutral atoms allows unique identification of the atomic states participating in the mutual neutralization process. Cross sections and branching ratios have also been calculated down to 1 meV center-of-mass collision energy for these two systems, with a multichannel Landau-Zener model and an asymptotic method for the ionic-covalent coupling matrix elements. The importance of two-electron processes in one-electron transfer is demonstrated by the dominant contribution of a core-excited configuration of the nitrogen atom in N+ + O- collisions. This effect is partially accounted for by introducing configuration mixing in the evaluation of coupling matrix elements.