We propose a novel dual-emission random Brillouin fiber laser (RBFL) with bidirectional pumping operation. Numerical simulations and experimental verification of the chaotic temporal and statistical properties of the RBFL are conducted, revealing intrinsic unpredictable intensity fluctuations and two completely uncorrelated laser outputs. A random bit generator based on quantum noise sources in the random Fabry-Perot resonator of the RBFL is realized at a bit rate of 5 Mbps with verified randomness.