Recent neutron scattering measurements of the spin-wave spectrum have shown that the second-nearest-neighbor exchange constant in EuO is ferromagnetic, in disagreement with previously published results from both specific-heat and magnetization measurements. We undertook a thorough study of the bulk data on both EuO and its isomorph EuS, including some previously unpublished specific-heat data. The new analysis resolved the controversy regarding the specific heat, which is actually in good agreement with the neutron scattering results. However, the NMR data are more sensitive to effects other than the exchange energy, making this technique inferior to the two other techniques regarding determination of the exchange constants. We obtained a set of "recommended" values for the nearest- and next-nearest-neighbor exchange constants in EuO and EuS. For EuO they are J1/k=0.606±0.008 and J2/k=0.119±0.015; and for EuS they are J1/k=0.228±0.004 and J2/k=-0.102±0.005.