We have measured the angular distributions of 1s photoelectrons excited by circularly and linearly polarized light from fixed-in-space CO and N2 molecules, in the vicinity of their shape resonances. A strong circular dichroism, i.e., a strong dependence on the sense of rotation of the polarization vector of the photons, is found for both molecules. State-of-the-art one-electron multiple scattering and partially correlated random phase approximation calculations are in good agreement with many, but not all, aspects of the experimental data.