  Standard sirens have been proposed as probes of alternative theories of
gravity, such as Horndeski models. Hitherto, all studies have been conducted on
a homogeneous-isotropic cosmological background, which is unable to
consistently account for realistic distributions of matter, and for
inhomogeneities in the Horndeski scalar field. Yet, the latter are essential
for screening mechanisms. In this article, we comprehensively analyze the
propagation of Horndeski gravitational waves in an arbitrary background
spacetime and scalar field. We restrict to the class of theories in which
gravitational waves propagate at light speed, and work in the geometric-optics
regime. We find that kinetic braiding only produces a nonphysical longitudinal
mode, whereas conformal coupling affects the amplitude of the standard
transverse modes but not their polarization. We confirm that any observable
deviation from general relativity depends on the local value of the effective
Planck mass at emission and reception of the wave. This result is interpreted
as the conservation of the number of gravitons.
