  We prove that if two free p.m.p. $\mathbb{Z}$-actions are Shannon orbit
equivalent then they have the same entropy. The argument also applies more
generally to yield the same conclusion for free p.m.p. actions of finitely
generated virtually Abelian groups. Together with the isomorphism theorems of
Ornstein and Ornstein-Weiss and the entropy invariance results of Austin and
Kerr-Li in the non-virtually-cyclic setting, this shows that two Bernoulli
actions of any non-locally-finite countably infinite amenable group are Shannon
orbit equivalent if and only if they are measure conjugate. We also show, at
the opposite end of the stochastic spectrum, that every $\mathbb{Z}$-odometer
is Shannon orbit equivalent to the universal $\mathbb{Z}$-odometer.
