Optical binding along the axis of two counterpropagating laser fields may be used to organize microparticles into longitudinal, spatially separated, arrays. Here we investigate correlations between the displacements of two optically bound microparticles from their equilibrium positions due to noise. Measurement of the decay time of the correlation functions of the center of mass and relative normal modes is shown to provide an in situ method to determine the optical restoring forces acting on the bound particles, thereby providing a test of our physical understanding of longitudinal optical binding.