  We develop data processing inequalities that describe how Fisher information
from statistical samples can scale with the privacy parameter $\varepsilon$
under local differential privacy constraints. These bounds are valid under
general conditions on the distribution of the score of the statistical model,
and they elucidate under which conditions the dependence on $\varepsilon$ is
linear, quadratic, or exponential. We show how these inequalities imply order
optimal lower bounds for private estimation for both the Gaussian location
model and discrete distribution estimation for all levels of privacy
$\varepsilon>0$. We further apply these inequalities to sparse Bernoulli models
and demonstrate privacy mechanisms and estimators with order-matching squared
$\ell^2$ error.
