Small-strain stiffness and damping ratio are the key parameters for modeling the dynamic behavior of soils, but they are often problematic and onerous to obtain under field conditions. This is particularly true for the attenuation characteristics. As an alternative to classical laboratory tests, that are costly and prone to inaccuracies due to scale and sampling issues, recent studies stress the opportunity of using in situ measurements, and particularly surface-wave methods. However, while these methods are often used to provide shear wave velocities, they are more rarely used to give in situ estimates of damping parameters. Similarly, it is uncommon to extract attenuation information from down-hole tests. In practice, the usual anti-earthquake design is confronted with a scarcity of laboratory test measurements and with the intricacies of obtaining reliable in situ attenuation measurements. Consequently, shaking simulations commonly use literature values of the damping coefficient. This choice is often motivated by the assumption that damping does not have a critical role in ground shaking evaluations, especially for weak-moderate motion cases. The purpose of this work is to provide, for these conditions, an analysis of when an accurate characterization of the damping coefficient is needed to produce reliable shaking predictions. We performed a stochastic analysis of synthetic seismic response for different realistic subsoil conditions using as input several real ground motion records. For completeness, we also compared linear-equivalent to fully non-linear shaking simulations, showing that, for the moderate-strain conditions we consider, the linear-equivalent approach is adequate. In our simulations we fixed the subsoil structures and material deformation properties while we adopted a Monte Carlo approach to explore the effects of randomly variable damping coefficient values. Our results show how an accurate knowledge of the attenuation coefficient is of minor importance in absence of strong impedance contrasts in the shallow subsurface.