  A powerful operational paradigm for distributed quantum information
processing involves manipulating pre-shared entanglement by local operations
and classical communication (LOCC). The LOCC round complexity of a given task
describes how many rounds of classical communication are needed to complete the
task. Despite some results separating one-round versus two-round protocols,
very little is known about higher round complexities. In this paper, we revisit
the task of one-shot random-party entanglement distillation as a way to
highlight some interesting features of LOCC round complexity. We first show
that for random-party distillation in three qubits, the number of communication
rounds needed in an optimal protocol depends on the entanglement measure used;
for the same fixed state some entanglement measures need only two rounds to
maximize whereas others need an unbounded number of rounds. In doing so, we
construct a family of LOCC instruments that require an unbounded number of
rounds to implement. We then prove explicit tight lower bounds on the LOCC
round number as a function of distillation success probability. Our
calculations show that the original W-state random distillation protocol by
Fortescue and Lo is essentially optimal in terms of round complexity.
