  We study the fundamental problem of frequency estimation under both privacy
and communication constraints, where the data is distributed among $k$ parties.
We consider two application scenarios: (1) one-shot, where the data is static
and the aggregator conducts a one-time computation; and (2) streaming, where
each party receives a stream of items over time and the aggregator continuously
monitors the frequencies. We adopt the model of multiparty differential privacy
(MDP), which is more general than local differential privacy (LDP) and
(centralized) differential privacy. Our protocols achieve optimality (up to
logarithmic factors) permissible by the more stringent of the two constraints.
In particular, when specialized to the $\varepsilon$-LDP model, our protocol
achieves an error of $\sqrt{k}/(e^{\Theta(\varepsilon)}-1)$ for all
$\varepsilon$, while the previous protocol (Chen et al., 2020) has error
$O(\sqrt{k}/\min\{\varepsilon, \sqrt{\varepsilon}\})$.
