We make an experimental assessment of the size of the 2D visibility complex of disjoint unit discs randomly distributed in the plane with density $\mu$. We observe that the number of free bitangents is asymptotically linear in the number of discs and we study the dependence of the linear asymptote in terms of the density of the scene. Specifically, for a particular range of scene densities $\mu$, we exhibit an approximation of the number of free bitangents in terms of $\mu$ and the number $n$ of discs, for $n$ larger than some function of $\mu$. We also notice how our approximation gained for rather large densities can be used to guess the onset of the linear behavior for small densities.