  We present the results of a set of cosmological N-body simulations with
standard $\Lambda$CDM cosmology but characterized by a scale-dependent
primordial non-Gaussianity of the local type featuring a power-law dependence
of the $f_{\rm NL}^{\rm loc}(k)$ at large scales followed by a saturation to a
constant value at smaller scales where non-linear growth leads to the formation
of collapsed cosmic structures. Such models are built to ensure consistency
with current Cosmic Microwave Background bounds on primordial non-Gaussianity
yet allowing for large effects of the non-Gaussian statistics on the properties
of non-linear structure formation. We show the impact of such scale-dependent
non-Gaussian scenarios on a wide range of properties of the resulting cosmic
structures, such as the non-linear matter power spectrum, the halo and sub-halo
mass functions, the concentration-mass relation, the halo and void density
profiles, and we highlight for the first time that some of these models might
mimic the effects of Warm Dark Matter for several of such observables
