  The interaction between a light mode and a mechanical oscillator via
radiation pressure in optomechanical systems is an excellent platform for a
multitude of applications in quantum technologies. In this work we study the
dynamics of a pair of optomechanical systems interacting dissipatively with a
wave guide in a unidirectional way. Focusing on the regime where the cavity
modes can be adiabatically eliminated we derive an effective coupling between
the two mechanical modes and we explore both classical and quantum correlations
established between the modes in both in the transient and in the stationary
regime, highlighting their asymmmetrical nature due to the unidirectional
coupling, and we find that a constant amount of steady correlations can exist
at long times. Furthermore we show that this unidirectional coupling
establishes a temperature gradient between the mirrors, depending on the
frequencies' detuning. We additionally analyze the power spectrum of the output
guide field and we show how, thanks to the chiral coupling, from such spectrum
it is possible to reconstruct the spectra of each single mirror.
