For decades transition-metal oxides have generated a huge interest due to the multitude of physical phenomena they exhibit. In this class of materials, the rare-earth nickelates, RNiO3, stand out for their rich phase diagram stemming from complex couplings between the lattice, electronic, and magnetic degrees of freedom. Here, we present a first-principles study of the low-temperature phase for two members of the RNiO3 series, with R = Pr, Y. We employ density-functional theory with Hubbard corrections accounting not only for the onsite localizing interactions among the Ni-3d electrons (U), but also the intersite hybridization effects between the transition metals and the ligands (V ). All the U and V parameters are calculated from first principles using densityfunctional perturbation theory, resulting in a fully ab initio methodology. Our simulations show that the inclusion of the intersite interaction parameters V is necessary to simultaneously capture the features well-established by experimental characterizations of the low-temperature state: insulating character, antiferromagnetism, and bond disproportionation. On the contrary, for some magnetic orderings the inclusion of onsite interaction parameters U alone completely suppresses the breathing distortion occurring in the low-temperature phase and produces an erroneous electronic state with a vanishing band gap. In addition-only when both the U and V are considered-we predict a polar phase with a magnetization-dependent electric polarization, supporting recent experimental observations that suggest a possible occurrence of type-II multiferroicity for these materials.