The problem of estimating the Poisson source localization using the inhomogeneous Poisson process observations results is considered. It is presupposed that k sensors are placed on the plane and each sensor processes the realization of the Poisson process with the intensity function depending upon the location of the source. A new mathematical theory for describing the asymptotic properties of the Bayesian and maximum likelihood estimates of the source localization is proposed. Special attention is paid to the analysis of the properties of the specified estimates depending on the regularity of the received signal front. In particular, the cases are considered when the intensity functions may be regular or may have cusp- or change-point-type singularities while their amplitudes are large. It is shown that, under the regularity conditions, the specified estimates are consistent, asymptotically normal and asymptotically efficient in terms of the minimax mean-square error. At the same time, in singular cases, only the Bayesian estimate is the effective one. Finally, some ways of implementation of the technically simple and consistent estimates of the Poisson source localization are also presented.