We study a full-duplex (FD) cell-free massive MIMO system where distributed access points (APs) operate in FD mode while user equipments (UEs) remain half-duplex. Although simultaneous uplink (UL) and downlink (DL) transmissions improve spectral efficiency, they introduce residual self-interference, AP-to-AP coupling, and UE-to-UE cross-link interference. Building on prior over-the-air distributed beamforming frameworks, we develop a fully distributed beamforming design based on iterative UL and DL pilot signaling under a joint UL and DL sum mean-square error criterion that explicitly accounts for these interference components. In FD operation, simultaneous UL and DL pilot transmissions cause UE-to-UE pilot leakage, which contaminates the reconstruction of the cross terms required for AP-specific beamforming design. To mitigate this effect, we introduce a pilot-domain projection of the received signals at the UEs, which suppresses the interference component and enables accurate cross-term reconstruction at the APs. In addition, best-response updates at the UEs are employed within the alternating optimization framework to improve convergence under strong UE-to-UE interference. Numerical results demonstrate faster convergence and higher effective sum rate, with particularly significant gains for strongly interfering UEs, compared with both separate UL and DL distributed OTA beamforming training schemes and designs based solely on local channel state information.