  We study a distributed sampling problem where a set of processors want to
output (approximately) independent and identically distributed samples from a
joint distribution with the help of a common message from a coordinator. Each
processor has access to a subset of sources from a set of independent sources
of "shared" randomness. We consider two cases -- in the "omniscient coordinator
setting", the coordinator has access to all these sources of shared randomness,
while in the "oblivious coordinator setting", it has access to none. All
processors and the coordinator may privately randomize. In the omniscient
coordinator setting, when the subsets at the processors are disjoint
(individually shared randomness model), we characterize the rate of
communication required from the coordinator to the processors over a multicast
link. For the two-processor case, the optimal rate matches a special case of
relaxed Wyner's common information proposed by Gastpar and Sula (2019), thereby
providing an operational meaning to the latter. We also give an upper bound on
the communication rate for the "randomness-on-the-forehead" model where each
processor observes all but one source of randomness and we give an achievable
strategy for the general case where the processors have access to arbitrary
subsets of sources of randomness. Also, we consider a more general model where
the processors observe components of correlated sources (with the coordinator
observing all the components), where we characterize the communication rate
when all the processors wish to output the same random sequence. In the
oblivious coordinator setting, we completely characterize the trade-off region
between the communication and shared randomness rates for the general case
where the processors have access to arbitrary subsets of sources of randomness.
