We investigate theoretically the intensity and phase correlations in the qu
antum-noise-driven dynamics of two spatially coupled microchip lasers. Thes
e correlations induce noise reduction phenomena for which simple analytic e
xpressions are derived. We predict a complete suppression of the dominant r
elaxation oscillation peaks in the intensity difference noise spectrum, as
well as photon statistics approaching the standard quantum limit. The corre
lations and consequent noise reduction effects are robust against frequency
detuning and persist for pump powers well above threshold.