  We develop lower bounds for estimation under local privacy
constraints---including differential privacy and its relaxations to approximate
or R\'{e}nyi differential privacy---by showing an equivalence between private
estimation and communication-restricted estimation problems. Our results apply
to arbitrarily interactive privacy mechanisms, and they also give sharp lower
bounds for all levels of differential privacy protections, that is, privacy
mechanisms with privacy levels $\varepsilon \in [0, \infty)$. As a particular
consequence of our results, we show that the minimax mean-squared error for
estimating the mean of a bounded or Gaussian random vector in $d$ dimensions
scales as $\frac{d}{n} \cdot \frac{d}{ \min\{\varepsilon, \varepsilon^2\}}$.
