  We explore the manifestation of quantum fluctuations across the
superconductor-insulator transition (SIT) in three different local
measurements: the two-particle density of states $P(\mathbf{r},\omega)$,
compressibility $\kappa(\mathbf{r})$, and diamagnetic susceptibility
$\chi(\mathbf{r})$. We perform Monte Carlo simulations on the 2D quantum
Josephson junction array model and present local maps of these quantities as
the system is tuned across the SIT. $P(\mathbf{r},\omega)$, obtained using
Maximum Entropy analytic continuation techniques, shows strongly diminished
zero-energy spectral weight in nearly-insulating islands, that also correlate
with regions of suppressed $\kappa(\mathbf{r})$. We investigate the signatures
of quantum fluctuations in the evolution of $\kappa(\mathbf{r})$ and
$\chi(\mathbf{r})$ across the SIT, distinct from thermal fluctuations. We
discuss the experimental implications of our results for scanning Josephson
spectroscopy, compressibility, and scanning SQUID measurements.
