  We develop an inhomogeneous quantum mean-field approach to the behavior of
collective excitations across the superfluid-Mott glass quantum phase
transition in two dimensions, complementing recent quantum Monte Carlo
simulations [Phys. Rev. Lett. {\bf 125}, 027002 (2020)]. In quadratic
(Gaussian) approximation, the Goldstone (phase) and Higgs (amplitude) modes
completely decouple. Each is described by a disordered Bogoliubov Hamiltonian
which can be solved by an inhomogeneous multi-mode Bogoliubov transformation.
We find that the Higgs mode is spatially localized in both phases. The
corresponding scalar spectral function shows a broad peak that is noncritical
in the sense that its peak frequency does not soften but remains nonzero across
the quantum phase transition. In contrast, the lowest-energy Goldstone mode
delocalizes in the superfluid phase, leading to a zero-frequency spectral peak.
We compare these findings to the results of the quantum Monte Carlo
simulations. We also relate them to general results on the localization of
bosonic excitations, and we discuss the limits and generality of our approach.
