  Gravitational waves can probe fundamental physics, leading to new constraints
on the mass of the graviton. Previous tests, however, have neglected the effect
of screening, which is typically present in modified theories that predict a
non-zero graviton mass. We here study whether future gravitational wave
observations can constrain the graviton mass when screening effects are taken
into account. We first consider model-independent corrections to the
propagation of gravitational waves due to screened massive graviton effects. We
find that future observation can place constraints on the screening radius and
the graviton mass of ${\cal{O}}(10^{2})$--${\cal{O}}(10^{4})$ Mpc and
${\cal{O}}(10^{-22})$--${\cal{O}}(10^{-26})$ eV respectively. We also consider
screening effects in two specific theories, ghost-free massive gravity and
bigravity, that might not realize these types of propagation modifications, but
that do provide analytic expressions for screening parameters relevant to our
analysis allowing for more concrete results. However, the constraints we are
able to place are small. The reason for this is that second- and
third-generation detectors are sensitive to graviton masses that lead to very
small screening radii in these particular models. The effect of screening,
however, can become important as constraints on the graviton mass are improved
through the stacking of multiple observations in the near future.
