  We study the cosmology of a specific class of nonlocal model of modified
gravity, the so-called Deser-Woodard (DW) model, modifying the Einstein-Hilbert
action by a term $\sim R f(\Box^{-1}R)$, where $f$ is a free function. Choosing
$f$ so as to reproduce the $\Lambda{\rm CDM}$ cosmological background expansion
history within the nonlocal model, we implement the model in a cosmological
linear Einstein--Boltzmann solver and study the deviations to GR the model
induces in the scalar and tensor perturbations. We observe that the DW nonlocal
model describes a modified propagation for the gravitational waves, as well as
a lower linear growth rate and a stronger lensing power as compared to
$\Lambda{\rm CDM}$, up to several percents. Such prominent growth and lensing
features lead to the inference of a significantly smaller value of $\sigma_8$
with respect to the one in $\Lambda{\rm CDM}$, given \textit{Planck}
CMB+lensing data. The prediction for the linear growth rate $f \sigma_8$ within
the DW model is therefore significantly smaller than the one in $\Lambda{\rm
CDM}$ and the addition of growth rate data $f \sigma_8$ from Redshift-space
distortion measurements to \textit{Planck} CMB+lensing, opens a (dominant)
tension between Redshift-space distortion data and the reconstructed
\textit{Planck} CMB lensing potential. However, model selection issues only
result in "weak" evidences for $\Lambda{\rm CDM}$ against the DW model given
the data. Such a fact shows that the datasets we consider are not constraining
enough for distinguishing between the models. As we discuss, the addition of
galaxy WL data or cosmological constraints from future galaxy clustering, weak
lensing surveys, but also third generation gravitational wave interferometers,
prove to be useful for discriminating modified gravity models such as the DW
one from $\Lambda{\rm CDM}$, within the close future.
