  We analyze the noise properties of two single electron transistors (SETs)
coupled via a shared voltage gate consisting of a nanomechanical resonator.
Working in the regime where the resonator can be treated as a classical system,
we find that the SETs act on the resonator like two independent heat baths. The
coupling to the resonator generates positive correlations in the currents
flowing through each of the SETs as well as between the two currents. In the
regime where the dynamics of the resonator is dominated by the back-action of
the SETs, these positive correlations can lead to parametrically large
enhancements of the low frequency current noise. These noise properties can be
understood in terms of the effects on the SET currents of fluctuations in the
state of a resonator in thermal equilibrium which persist for times of order
the resonator damping time.
