It is shown that a gravitationally bound system with a one-dimensional
velocity dispersion sigma can at most dissipate a fraction ~36(sigma/c)^3 of
the gravitational wave energy propagating through it. The limit is saturated
for particles with a mean-free-path equal to the size of the system, such as
hot protons in galaxy clusters, strongly-interacting dark matter particles in
halos, or massive black holes in clusters. Even for these systems, the
dissipated fraction, <10^{-6}, does not degrade the use of gravitational waves
as standard sirens for cosmology. In particular, the dissipation effect is
smaller by a factor ~0.03(c/sigma)^2 than the peculiar velocity uncertainty of
the source within its host dynamical system. The electromagnetic counterparts
that result from the dissipation are too faint to be detectable at cosmological
distances.