  We study the problem of estimating a functional $\theta(\mathbb P)$ of an
unknown probability distribution $\mathbb P \in\mathcal P$ in which the
original iid sample $X_1,\dots, X_n$ is kept private even from the statistician
via an $\alpha$-local differential privacy constraint. Let $\omega_{TV}$ denote
the modulus of continuity of the functional $\theta$ over $\mathcal P$, with
respect to total variation distance. For a large class of loss functions $l$
and a fixed privacy level $\alpha$, we prove that the privatized minimax risk
is equivalent to $l(\omega_{TV}(n^{-1/2}))$ to within constants, under
regularity conditions that are satisfied, in particular, if $\theta$ is linear
and $\mathcal P$ is convex. Our results complement the theory developed by
Donoho and Liu (1991) with the nowadays highly relevant case of privatized
data. Somewhat surprisingly, the difficulty of the estimation problem in the
private case is characterized by $\omega_{TV}$, whereas, it is characterized by
the Hellinger modulus of continuity if the original data $X_1,\dots, X_n$ are
available. We also find that for locally private estimation of linear
functionals over a convex model a simple sample mean estimator, based on
independently and binary privatized observations, always achieves the minimax
rate. We further provide a general recipe for choosing the functional parameter
in the optimal binary privatization mechanisms and illustrate the general
theory in numerous examples. Our theory allows to quantify the price to be paid
for local differential privacy in a large class of estimation problems. This
price appears to be highly problem specific.
