Calculations of the 16O(e,e′p N) cross sections to the ground state and first excited levels of the 14C and 14N nuclei are presented. The effects of nuclear fragmentation have been obtained in a self-consistent approach and are accounted for in the determination of the two-nucleon removal amplitudes. The Hilbert space is partitioned in order to compute the contribution of both long- and short-range effects in a separate way. Both the two-proton and the proton-neutron emission cross sections have been computed within the same model for the nuclear structure as well as the same treatment of the reaction mechanism, with the aim of better comparing the differences between the two physical processes. The 16O(e,e′ pp) reaction is found to be sensitive to short-range correlations, in agreement with previous results. The 16O(e,e′ pn) cross section to 1+ final states is dominated by the Δ current and tensor correlations. For both reactions, the interplay between collective (long-range) effects and short-range and tensor correlations plays an important role. This suggests that the selectivity of (e,e′pN) reactions to the final state can be used to probe correlations also beyond short-range effects.