Spin-torque effects in antiferromagnetic (AFM) materials are of great interest due to the possible applications as high-speed spintronic devices. In the present paper we analyze the statistical properties of the current-driven AFM nanooscillator that result from the white Gaussian noise of magnetic nature. According to the peculiarities of deterministic dynamics, we derive the Langevin and Fokker-Planck equations in the energy representation of two normal modes. We find the stationary distribution function in the subcritical and overcritical regimes and calculate the current dependence of the average energy, energy fluctuation and their ratio (quality factor). The noncritical mode shows the Boltzmann statistics with the current-dependent effective temperature in the whole range of the current values. The effective temperature of the other, i.e., soft, mode critically depends on the current in the subcritical region. Distribution function of the soft mode follows the Gaussian law above the generation threshold. In the overcritical regime, the total average energy and the quality factor grow with the current value. This raises the AFM nanooscillators to the promising candidates for active spintronic components.