We present theory and computational method for analyzing dissociative resonant photoemission from first principles. Particular emphasis is devoted to the conditions for observing so-called atomic peaks and atomic holes. The atomic peaks are connected with photoemission following resonant excitation to dissociative core excited states which show signals from scattering channels involving the dissociation (atomic) fragments in addition to those involving the compound molecule. The holes are the results of continuum-continuum interference effects between these two, atomic and molecular, channels which may act destructively under certain conditions. We apply a novel electronic structure method to compute the transition moments for the resonant and direct photoemission channels including their dependence on internuclear distances and their interference. The relevant matrix elements involving the photoelectron are obtained using similar techniques for the two types of channels, with the scattered electron wave in each case being determined in the full molecular anisotropic potential. A study of resonant photoemission through the core excited sigma* states of HF and HCl indicates that the appearance of the atomic peaks and holes is subtly dependent on the nuclear dynamics, the potential energy curves, and the excitation photon frequency. We demonstrate that the resonant contribution and the evolution of the atomic peaks can be subject to strong dynamical suppression, so strong in fact that main state photoionization may constitute the dominating channel even at resonant conditions. It is shown that such dynamical suppression explains that resonant excitation to the F 1s-sigma* dissociative state in hydrogen fluoride gives a photoelectron spectrum in which the spectator part contains strong atomic lines but a participator part where such lines are lacking, although they both refer to the same, dissociative, core excited state. The findings in the present work give evidence that both direct and resonant channels should be simultaneously considered in analyses of the dissociative photoemission process even at resonant conditions.