  Independent samples from an unknown probability distribution $\bf p$ on a
domain of size $k$ are distributed across $n$ players, with each player holding
one sample. Each player can communicate $\ell$ bits to a central referee in a
simultaneous message passing model of communication to help the referee infer a
property of the unknown $\bf p$. What is the least number of players for
inference required in the communication-starved setting of $\ell<\log k$? We
begin by exploring a general "simulate-and-infer" strategy for such inference
problems where the center simulates the desired number of samples from the
unknown distribution and applies standard inference algorithms for the
collocated setting. Our first result shows that for $\ell<\log k$ perfect
simulation of even a single sample is not possible. Nonetheless, we present a
Las Vegas algorithm that simulates a single sample from the unknown
distribution using $O(k/2^\ell)$ samples in expectation. As an immediate
corollary, we get that simulate-and-infer attains the optimal sample complexity
of $\Theta(k^2/2^\ell\epsilon^2)$ for learning the unknown distribution to
total variation distance $\epsilon$. For the prototypical testing problem of
identity testing, simulate-and-infer works with $O(k^{3/2}/2^\ell\epsilon^2)$
samples, a requirement that seems to be inherent for all communication
protocols not using any additional resources. Interestingly, we can break this
barrier using public coins. Specifically, we exhibit a public-coin
communication protocol that performs identity testing using
$O(k/\sqrt{2^\ell}\epsilon^2)$ samples. Furthermore, we show that this is
optimal up to constant factors. Our theoretically sample-optimal protocol is
easy to implement in practice. Our proof of lower bound entails showing a
contraction in $\chi^2$ distance of product distributions due to communication
constraints and may be of independent interest.
