  A growing number of gamma-ray burst (GRB) afterglows is observed at very-high
energies (VHE, $\gtrsim 100$ GeV). Yet, our understanding of the mechanism
powering the VHE emission remains baffling. We make use of multi-wavelength
observations of the afterglow of GRB 180720B, GRB 190114C, and GRB 221009A to
investigate whether the bursts exhibiting VHE emission share common features.
We assume the standard afterglow model and microphysical parameters consistent
with a synchrotron self-Compton (SSC) scenario for the VHE radiation. By
requiring that the blastwave should be transparent to $\gamma$-$\gamma$ pair
production at the time of observation of the VHE photons and relying on typical
prompt emission efficiencies and data in the radio, optical and X-ray bands, we
infer for those bursts that the initial energy of the blastwave is
$\tilde{E}_{k, \rm{iso}} \gtrsim \mathcal{O}(10^{54})$ erg and the circumburst
density is $n_0 \lesssim \mathcal{O}(10^{-1})$ cm$^{-3}$ for a constant
circumburst profile [or $A_\star \lesssim \mathcal{O}(10^{-1})$ cm$^{-1}$ for a
wind scenario]. Our findings thus suggest that these VHE bursts might be hosted
in low-density environments, if the SSC radiation is responsible for the VHE
emission. While these trends are based on a small number of bursts, the
Cherenkov Telescope Array has the potential to provide crucial insight in this
context by detecting a larger sample of VHE GRBs. In addition, due to the very
poor statistics, the non-observation of high-energy neutrinos cannot constrain
the properties of these bursts efficiently, unless additional VHE GRBs should
be detected at distances closer than $15$ Mpc when IceCube-Gen2 radio will be
operational.
