This paper investigates the so-far unexplored influence of the probes on the state of polarization (SOP) of the pump pulse in a distributed fiber sensor based on dual-probe Brillouin optical-time domain analysis (BOTDA). A theoretical framework, based on a vector formalism in the Stokes space, is proposed to describe the Brillouin interaction of a dual-probe BOTDA setup. Experimental results over a 50 km long fiber, supported by numerical simulations over a few km of a modeled birefringent fiber, demonstrate for the first time notable Brillouinmediated attraction/repulsion actions from the probes on the SOP of the pump pulse. While the probes experience Brillouin-induced polarization transformation only during their relatively short interaction with the pump pulse, the effect of the probes on the pump SOP accumulates over the entire fiber length, becoming more significant with higher probe power. The opposite actions of the two (gain and loss) probes do not provide full compensation, since they are dissimilarly affected by the pump, and have slightly different associated Brillouin frequency shifts. The study highlights an overlooked limitation in BOTDA sensing, where the different influences of the probes on the pump SOP impair the commonly used method for polarization fading mitigation based on a polarization switch. This method involves launching a pair of orthogonally co-polarized probes (or pumps) propagating in the sensing fiber against a single-polarization pump (or singlepolarization dual-sideband probes). This limitation becomes particularly relevant in advanced configurations using high probe powers, especially for long sensing ranges and high-performance applications.