We address the long-standing question of the nature of oxygen vacancies in strontium titanate, using a combination of density functional theory and dynamical mean-field theory (DFT+DMFT) to investigate in particular the effect of vacancy-site correlations on the electronic properties. Our approach uses a minimal low-energy electronic subspace including the Ti-t2g orbitals plus an additional vacancy-centered Wannier function, and it provides an intuitive and physically transparent framework to study the effect of the local electron-electron interactions on the excess charge introduced by the oxygen vacancies. We estimate the strength of the screened interaction parameters using the constrained random phase approximation, and we find a sizable Hubbard U parameter for the vacancy orbital. Our main finding, which reconciles previous experimental and computational results, is that the ground state is either a state with double occupation of the localized defect state or a state with a singly occupied vacancy and one electron transferred to the conduction band. The balance between these two competing states is determined by the strength of the interaction both on the vacancy and the Ti sites, and on the Ti-Ti distance across the vacancy. Finally, we contrast the case of vacancy doping in SrTiO3 with doping via La substitution, and we show that the latter is well described by a simple rigid-band picture.