  We present a novel way to compare the statistical cost of privacy mechanisms
using their Dobrushin coefficient. Specifically, we provide upper and lower
bounds for the Dobrushin coefficient of a privacy mechanism in terms of its
maximal leakage and local differential privacy guarantees. Given the geometric
nature of the Dobrushin coefficient, this approach provides some insights into
the general statistical cost of these privacy guarantees. We highlight the
strength of this method by applying our results to develop new bounds on the
$\ell_2$-minimax risk in a distribution estimation setting under maximal
leakage constraints. Specifically, we find its order with respect to the sample
size and privacy level, allowing a quantitative comparison with the
corresponding local differential privacy $\ell_2$-minimax risk.
