  Recent studies of galaxies in the local Universe, including those in the
Local Group, find that the efficiency of environmental (or satellite) quenching
increases dramatically at satellite stellar masses below ~ $10^8\ {\rm
M}_{\odot}$. This suggests a physical scale where quenching transitions from a
slow "starvation" mode to a rapid "stripping" mode at low masses. We
investigate the plausibility of this scenario using observed HI surface density
profiles for a sample of 66 nearby galaxies as inputs to analytic calculations
of ram-pressure and viscous stripping. Across a broad range of host properties,
we find that stripping becomes increasingly effective at $M_{*} < 10^{8-9}\
{\rm M}_{\odot}$, reproducing the critical mass scale observed. However, for
canonical values of the circumgalactic medium density ($n_{\rm halo} <
10^{-3.5}$ ${\rm cm}^{-3}$), we find that stripping is not fully effective;
infalling satellites are, on average, stripped of < 40 - 70% of their cold gas
reservoir, which is insufficient to match observations. By including a host
halo gas distribution that is clumpy and therefore contains regions of higher
density, we are able to reproduce the observed HI gas fractions (and thus the
high quenched fraction and short quenching timescale) of Local Group
satellites, suggesting that a host halo with clumpy gas may be crucial for
quenching low-mass systems in Local Group-like (and more massive) host halos.
