  In this work, we extend previous analyses of the structure formation in the
$f(\Box^{-1}R)$ model of nonlocal gravity proposed by Deser and Woodard (DW),
which reproduces the background expansion of $\Lambda$CDM with no need of a
cosmological constant nor of any dimensional constant beside Newton's one. A
previous analysis based on redshift-space distortions (RSD) data concluded that
the model was ruled out. In this work we revisit the issue and find that, when
recast in a localized model, the DW model is not ruled out and actually gives a
better fit to RSD data than $\Lambda$CDM. At the same time, the model predicts
a slightly lower value of $\sigma_{8}$ than $\Lambda$CDM, in agreement with
recent estimates based on lensing. We also produce analytical approximations of
the two modified gravity functions and of $f\sigma_{8}(z)$ as a function of
redshift. Finally, we also show how much the fit depends on initial conditions
when these are generalized with respect to a standard matter-dominated era.
