  We consider the effect of a time-varying Planck mass on the propagation of
gravitational waves (GWs). A running Planck mass arises naturally in several
modified gravity theories, and here we focus on those that carry an additional
dark energy field responsible for the late-time accelerated expansion of the
universe, yet--like general relativity (GR)--propagate only two GW
polarizations, both traveling at the speed of light. Because a time-varying
Planck mass affects the amplitude of the GWs and therefore the inferred
distance to the source, standard siren measurements of $H_0$ are degenerate
with the parameter $c_M$ characterizing the time-varying Planck mass, where
$c_M=0$ corresponds to GR with a constant Planck mass. The effect of non-zero
$c_M$ will have a noticeable impact on GWs emitted by binary neutron stars
(BNSs) at the sensitivities and distances observable by ground-based GW
detectors such as advanced LIGO and A+, implying that standard siren
measurements can provide joint constraints on $H_0$ and $c_M$. Assuming a
$\Lambda$CDM evolution of the universe and taking Planck's measurement of $H_0$
as a prior, we find that GW170817 constrains $c_M = -9^{+21}_{-28}$ ($68.3\%$
credibility). We also discuss forecasts, finding that if we assume $H_0$ is
known independently, then 100 BNS events detected by advanced LIGO can
constrain $c_M$ to within $\pm0.9$. This is comparable to the current best
constraints from cosmology. Similarly, for 100 LIGO A+ BNS detections, it is
possible to constrain $c_M$ to $\pm0.5$. When analyzing joint $H_0$ and $c_M$
constraints we find that $\sim 400$ LIGO A+ events are needed to constrain
$H_0$ to $1\%$ accuracy. Finally, we discuss the possibility of a nonzero value
of $c_M$ biasing standard siren $H_0$ measurements from 100 LIGO A+ detections,
and find that $c_M=+1.35$ could bias $H_0$ by 3-4$\sigma$ too low if we
incorrectly assume $c_M=0$.
