  This work provides tight upper- and lower-bounds for the problem of mean
estimation under $\epsilon$-differential privacy in the local model, when the
input is composed of $n$ i.i.d. drawn samples from a normal distribution with
variance $\sigma$. Our algorithms result in a $(1-\beta)$-confidence interval
for the underlying distribution's mean $\mu$ of length $\tilde O\left(
\frac{\sigma \sqrt{\log(\frac 1 \beta)}}{\epsilon\sqrt n} \right)$. In
addition, our algorithms leverage binary search using local differential
privacy for quantile estimation, a result which may be of separate interest.
Moreover, we prove a matching lower-bound (up to poly-log factors), showing
that any one-shot (each individual is presented with a single query) local
differentially private algorithm must return an interval of length
$\Omega\left( \frac{\sigma\sqrt{\log(1/\beta)}}{\epsilon\sqrt{n}}\right)$.
