  Bell nonlocality can be formulated in terms of a resource theory with
local-hidden variable models as resourceless objects. Two such theories are
known, one built upon local operations assisted by shared randomness (LOSRs)
and the other one allowing, in addition, for prior-to-input classical
communication. We show that prior communication, although unable to create
nonlocality, leads to wirings not only beyond LOSRs but also not contained in a
much broader class of (nonlocality-generating) global wirings. Technically,
this is shown by proving that it can improve the statistical distinguishability
between Bell correlations optimised over all fixed measurement choices. This
has implications in nonlocality quantification, and leads us to a natural
universal definition of Bell nonlocality measures. To end up with, we also
consider the statistical strength of nonlocality proofs. We point out some
issues of its standard definition in the resource-theoretic operational
framework, and suggest simple fixes for them. Our findings reveal non-trivial
features of the geometry of the set of wirings and may have implications in the
operational distinguishability of nonlocal behaviors.
