  We consider a setup in which confidential i.i.d. samples $X_1,\dotsc,X_n$
from an unknown finite-support distribution $\boldsymbol{p}$ are passed through
$n$ copies of a discrete privatization channel (a.k.a. mechanism) producing
outputs $Y_1,\dotsc,Y_n$. The channel law guarantees a local differential
privacy of $\epsilon$. Subject to a prescribed privacy level $\epsilon$, the
optimal channel should be designed such that an estimate of the source
distribution based on the channel outputs $Y_1,\dotsc,Y_n$ converges as fast as
possible to the exact value $\boldsymbol{p}$. For this purpose we study the
convergence to zero of three distribution distance metrics: $f$-divergence,
mean-squared error and total variation. We derive the respective normalized
first-order terms of convergence (as $n\to\infty$), which for a given target
privacy $\epsilon$ represent a rule-of-thumb factor by which the sample size
must be augmented so as to achieve the same estimation accuracy as that of a
non-randomizing channel. We formulate the privacy-fidelity trade-off problem as
being that of minimizing said first-order term under a privacy constraint
$\epsilon$. We further identify a scalar quantity that captures the essence of
this trade-off, and prove bounds and data-processing inequalities on this
quantity. For some specific instances of the privacy-fidelity trade-off
problem, we derive inner and outer bounds on the optimal trade-off curve.
