  A central server needs to perform statistical inference based on samples that
are distributed over multiple users who can each send a message of limited
length to the center. We study problems of distribution learning and identity
testing in this distributed inference setting and examine the role of shared
randomness as a resource. We propose a general-purpose simulate-and-infer
strategy that uses only private-coin communication protocols and is
sample-optimal for distribution learning. This general strategy turns out to be
sample-optimal even for distribution testing among private-coin protocols.
Interestingly, we propose a public-coin protocol that outperforms
simulate-and-infer for distribution testing and is, in fact, sample-optimal.
Underlying our public-coin protocol is a random hash that when applied to the
samples minimally contracts the chi-squared distance of their distribution to
the uniform distribution.
