<p>We propose an on-shell description of spinning binary systems in gravitational theories where compact objects display scalar hair. The framework involves matter particles of arbitrary spin which, in addition to the minimal coupling to gravitons, are conformally coupled to a massless scalar mediating non-standard interactions. We use the unitary factorization techniques to derive the on-shell amplitudes relevant for emission of scalars and gravitons in matter scattering, paying attention to parametrize the freedom due to contact terms. Using the KMOC formalism, these amplitudes allow one to derive succinct expressions for the radiation waveforms at the leading post-Minkowskian order, together with the associated memory effects. Furthermore, we compute the power emitted via gravitational and scalar radiation in hyperbolic encounters of compact objects. After a continuation to bound orbits, these are compared with results obtained in specific scalar-tensor theories where black holes exhibit scalar hair, such as the scalar-Gauss-Bonnet or dynamical Chern-Simons theories. Finally, we identify possible deformations from the conformal coupling that can contribute to radiation observables at the same post-Newtonian order.</p>