We report on the relative length fluctuation of two fixed-spacer Fabry-Perot cavities with mirrors fabricated from silica/tantala dielectric coatings on fused silica substrates. By locking a laser to each cavity and reading out the beat note (nu) over cap = nu(1) - nu(2) of the transmitted beams, we find that, for frequencies from 10 Hz to 1 kHz, the power spectral density of beat note fluctuation is S-(nu) over cap (f)= (0.5Hz)(2)/f. By careful budgeting of noise sources contributing to the beat note, we find that our measurement is consistent with the fluctuation in this band being dominated by the Brownian noise of the mirror coatings. Fitting for the coating loss angle.c, we find it equal to 4 x 10(-4). We then use a Bayesian analysis to combine our measurement with previous observations, and thereby extract estimates for the individual loss angles of the silica and tantala constituents of these coatings. With minor upgrades, the testbed described in this article can be used in the future to measure the length noise of cavities formed with novel mirror coating materials and geometries.