  We study goodness-of-fit of discrete distributions in the distributed
setting, where samples are divided between multiple users who can only release
a limited amount of information about their samples due to various information
constraints. Recently, a subset of the authors showed that having access to a
common random seed (i.e., shared randomness) leads to a significant reduction
in the sample complexity of this problem. In this work, we provide a complete
understanding of the interplay between the amount of shared randomness
available, the stringency of information constraints, and the sample complexity
of the testing problem by characterizing a tight trade-off between these three
parameters. We provide a general distributed goodness-of-fit protocol that as a
function of the amount of shared randomness interpolates smoothly between the
private- and public-coin sample complexities. We complement our upper bound
with a general framework to prove lower bounds on the sample complexity of this
testing problems under limited shared randomness. Finally, we instantiate our
bounds for the two archetypal information constraints of communication and
local privacy, and show that our sample complexity bounds are optimal as a
function of all the parameters of the problem, including the amount of shared
randomness.
  A key component of our upper bounds is a new primitive of domain compression,
a tool that allows us to map distributions to a much smaller domain size while
preserving their pairwise distances, using a limited amount of randomness.
