  We study simple binary hypothesis testing under both local differential
privacy (LDP) and communication constraints. We qualify our results as either
minimax optimal or instance optimal: the former hold for the set of
distribution pairs with prescribed Hellinger divergence and total variation
distance, whereas the latter hold for specific distribution pairs. For the
sample complexity of simple hypothesis testing under pure LDP constraints, we
establish instance-optimal bounds for distributions with binary support;
minimax-optimal bounds for general distributions; and (approximately)
instance-optimal, computationally efficient algorithms for general
distributions. When both privacy and communication constraints are present, we
develop instance-optimal, computationally efficient algorithms that achieve the
minimum possible sample complexity (up to universal constants). Our results on
instance-optimal algorithms hinge on identifying the extreme points of the
joint range set $\mathcal A$ of two distributions $p$ and $q$, defined as
$\mathcal A := \{(\mathbf T p, \mathbf T q) | \mathbf T \in \mathcal C\}$,
where $\mathcal C$ is the set of channels characterizing the constraints.
