Reference proposed a flavor model based on the symmetry group S4, managing to describe fermion masses and mixings. The Weinberg operator has been used in order to provide the smallness of the neutrino masses, while a set of scalar fields, getting nonvanishing vacuum expectation values, spontaneously breaks down S4 and provides the tri-bimaximal pattern as the lepton mixing matrix. Restricting to this setting, in this paper we analyze possible origins for the effective terms: the type I seesaw mechanism is the best known approach, but type II and III are also discussed. The phenomenology related to these models is various and future experiments could in principle discriminate among these proposals. Furthermore, we compare our realizations to two relevant A4 based models, also predicting the tri-bimaximal lepton mixing, and we find that an analysis of the 0ν2β-decay parameters could distinguish among all these realizations. The introduction of new physics beyond the standard model, like heavy right-handed neutrinos, scalar triplets, and fermion triplets, let us investigate leptogenesis, and this provides constraints in the realization of the models. © 2009 The American Physical Society.