  We analyze the propagation of high-frequency gravitational waves (GW) in
scalar-tensor theories of gravity, with the aim of examining properties of
cosmological distances as inferred from GW measurements. By using symmetry
principles, we first determine the most general structure of the GW linearized
equations and of the GW energy momentum tensor, assuming that GW move with the
speed of light. Modified gravity effects are encoded in a small number of
parameters, and we study the conditions for ensuring graviton number
conservation in our covariant set-up. We then apply our general findings to the
case of GW propagating through a perturbed cosmological space-time, deriving
the expressions for the GW luminosity distance $d_L^{({\rm GW})}$ and the GW
angular distance $d_A^{({\rm GW})}$. We prove for the first time the validity
of Etherington reciprocity law $d_L^{({\rm GW})}\,=\,(1+z)^2\,d_A^{({\rm GW})}$
for a perturbed universe within a scalar-tensor framework. We find that besides
the GW luminosity distance, also the GW angular distance can be modified with
respect to General Relativity. We discuss implications of this result for
gravitational lensing, focussing on time-delays of lensed GW and lensed photons
emitted simultaneously during a multimessenger event. We explicitly show how
modified gravity effects compensate between different coefficients in the GW
time-delay formula: lensed GW arrive at the same time as their lensed
electromagnetic counterparts, in agreement with causality constraints.
