The interaction of random laser and gain medium is important to understand the noise origin in random fiber lasers. Here, using the optical time domain reflectometry method, the time-resolved distributed acoustic wave generated by a Brillouin random fiber laser (BRFL) is characterized. The dynamic property of the acoustic wave reflects the gain dynamics of the BRFL. The principle is based on the polarization-decoupled stimulated Brillouin scattering (SBS)-enhanced four-wave mixing process, where the probe light experiences maximum reflection when the phase match condition is satisfied. Static measurements present exponentially depleted Brillouin gain along the gain medium in the BRFL, indicating the localized random SBS frequency change in the maximum local gain region, which varies with time to contribute random laser noise as revealed in the dynamic measurement. The SBS-induced birefringence change in the Brillouin gain fiber is approximately 10-7 to 10-6. The phase noise of the BRFL is observed directly inside the random laser gain medium for the first time via time and spatially varied acoustic wave intensity. By counting the temporal intensity statistical distribution, optical rogue waves are detected near the lasing threshold of the BRFL. Different temporal intensity statistical distribution at high and low gain positions is found, which is caused by the SBS nonlinear transfer function and localized gain. The distributed characterization methods in the paper provide a new platform to study the interaction of random lasers and gain medium, giving us a new perspective to understand the fundamental physics of the random lasing process and its noise property.