  We investigate under which conditions quantum nonlocal manifestations as
Einstein-Podolsky-Rosen steering or Bell nonlocality can manifest themselves
even at the macroscopic level of two mechanical resonators in optomechanical
systems. We adopt the powerful scheme of reservoir engineering, implemented by
driving a cavity mode with a properly chosen two-tone field, to prepare two
mechanical oscillators into an entangled state. We show that large and robust
(both one-way and two-way) steering could be achieved in the steady state with
realistic parameters. We analyze the mechanism of the asymmetric nature of
steering in our system of two-mode Gaussian state. However, unlike steering,
Bell nonlocality is present under much more stringent conditions. We consider
two types of measurements, displaced parity and on-off detection, respectively.
We show that for both the measurements Bell violation requires very low
environmental temperature. For the parity detection, large Bell violation is
observed only in the transient state when the mechanical modes decouple from
the optical mode and with extremely small cavity losses and mechanical damping.
Whereas for the on-off detection, moderate Bell violation is found in the
steady state and robust against cavity losses and mechanical damping. Although
Bell violation with the parity detection seems extremely challenging to be
experimentally demonstrated, the conditions required for violating Bell
inequalities with the on-off detection are much less demanding.
