This paper studies the distributed optimal bilinear equalizer (OBE) beamforming design for both the uplink and downlink cell-free massive multiple-input multiple-output networks. We consider arbitrary statistics-based channel estimators over spatially correlated Rician fading channels. In the uplink, we derive the achievable spectral efficiency (SE) performance and OBE combining schemes with arbitrary statistics-based channel estimators and compute their respective closed-form expressions. It is insightful to explore that the achievable SE performance is not dependent on the choice of channel estimator when OBE combining schemes are applied over Rayleigh channels. In the downlink, we derive the achievable SE performance expressions with BE precoding schemes and arbitrary statistics-based channel estimators utilized and compute them in closed form. Then, we obtain the OBE precoding scheme leveraging insights from uplink OBE combining schemes.