  We study the nonlinear growth of structure in nonlocal gravity models with
the aid of N-body simulation and the spherical collapse and halo models. We
focus on a model in which the inverse-squared of the d'Alembertian operator
acts on the Ricci scalar in the action. For fixed cosmological parameters, this
model differs from $\Lambda{\rm CDM}$ by having a lower late-time expansion
rate and an enhanced and time-dependent gravitational strength ($\sim 6\%$
larger today). Compared to $\Lambda{\rm CDM}$ today, in the nonlocal model,
massive haloes are slightly more abundant (by $\sim 10\%$ at $M \sim 10^{14}
M_{\odot}/h$) and concentrated ($\approx 8\%$ enhancement over a range of mass
scales), but their linear bias remains almost unchanged. We find that the
Sheth-Tormen formalism describes the mass function and halo bias very well,
with little need for recalibration of free parameters. The fitting of the halo
concentrations is however essential to ensure the good performance of the halo
model on small scales. For $k \gtrsim 1 h/{\rm Mpc}$, the amplitude of the
nonlinear matter and velocity divergence power spectra exhibits a modest
enhancement of $\sim 12\%$ to $15\%$, compared to $\Lambda{\rm CDM}$ today.
This suggests that this model might only be distinguishable from $\Lambda{\rm
CDM}$ by future observational missions. We point out that the absence of a
screening mechanism may lead to tensions with Solar System tests due to local
time variations of the gravitational strength, although this is subject to
assumptions about the local time evolution of background averaged quantities.
