  In the \emph{shuffle model} of differential privacy, data-holding users send
randomized messages to a secure shuffler, the shuffler permutes the messages,
and the resulting collection of messages must be differentially private with
regard to user data. In the \emph{pan-private} model, an algorithm processes a
stream of data while maintaining an internal state that is differentially
private with regard to the stream data. We give evidence connecting these two
apparently different models.
  Our results focus on \emph{robustly} shuffle private protocols, whose privacy
guarantees are not greatly affected by malicious users. First, we give robustly
shuffle private protocols and upper bounds for counting distinct elements and
uniformity testing. Second, we use pan-private lower bounds to prove robustly
shuffle private lower bounds for both problems. Focusing on the dependence on
the domain size $k$, we find that robust approximate shuffle privacy and
approximate pan-privacy have additive error $\Theta(\sqrt{k})$ for counting
distinct elements. For uniformity testing, we give a robust approximate shuffle
private protocol with sample complexity $\tilde O(k^{2/3})$ and show that an
$\Omega(k^{2/3})$ dependence is necessary for any robust pure shuffle private
tester. Finally, we show that this connection is useful in both directions: we
give a pan-private adaptation of recent work on shuffle private histograms and
use it to recover further separations between pan-privacy and interactive local
privacy.
