  We examine the origin of two opposite results for the growth of perturbations
in the Deser-Woodard (DW) nonlocal gravity model. One group previously analyzed
the model in its original nonlocal form and showed that the growth of structure
in the DW model is enhanced compared to general relativity (GR) and thus
concluded that the model was ruled out. Recently, however, another group has
reanalyzed it by localizing the model and found that the growth in their
localized version is suppressed even compared to the one in GR. The question
was whether the discrepancy originates from an intrinsic difference between the
nonlocal and localized formulations or is due to their different
implementations of the sub-horizon limit. We show that the nonlocal and local
formulations give the same solutions for the linear perturbations as long as
the initial conditions are set the same. The different implementations of the
sub-horizon limit lead to different transient behaviors of some perturbation
variables; however, they do not affect the growth of matter perturbations at
the sub-horizon scale much. In the meantime, we also report an error in the
numerical calculation code of the former group and verify that after fixing the
error the nonlocal version also gives the suppressed growth. Finally, we
discuss two alternative definitions of the effective gravitational constant
taken by the two groups and some open problems.
