  We prove that the local eigenvalue statistics at energy $E$ in the
localization regime for Schr\"odinger operators with random point interactions
on $\mathbb{R}^d$, for $d=1,2,3$, is a Poisson point process with the intensity
measure given by the density of states at $E$ times the Lebesgue measure. This
is one of the first examples of Poisson eigenvalue statistics for the
localization regime of multi-dimensional random Schr\"odinger operators in the
continuum. The special structure of resolvent of Schr\"odinger operators with
point interactions facilitates the proof of the Minami estimate for these
models.
