  Neutrinoless $\beta\beta$ decay nuclear matrix elements calculated with the
shell model and energy-density functional theory typically disagree by more
than a factor of two in the standard scenario of light-neutrino exchange. In
contrast, for a decay mediated by sterile heavy neutrinos the deviations are
reduced to about 50\%, an uncertainty similar to the one due to short-range
effects. We compare matrix elements in the light- and heavy-neutrino-exchange
channels, exploring the radial, momentum transfer and angular momentum-parity
matrix element distributions, and considering transitions that involve
correlated and uncorrelated nuclear states. We argue that the shorter-range
heavy-neutrino exchange is less sensitive to collective nuclear correlations,
and that discrepancies in matrix elements are mostly due to the treatment of
long-range correlations in many-body calculations. Our analysis supports
previous studies suggesting that isoscalar pairing correlations, which affect
mostly the longer-range part of the neutrinoless $\beta\beta$ decay operator,
are partially responsible for the differences between nuclear matrix elements
in the standard light-neutrino-exchange mechanism.
