The Kalman filter (KF) and Maximum a Posteriori (MAP) methods are two types of generalized Bayesian filters in structural load identification. KF is computationally efficient with poor numerical stability, and in contrast, MAP is highly applicable requiring complex matrix inversion operations. Additionally, both methods have limitations on the load distribution and sensor arrangement, applying to simple types of loads. To address these challenges, a Bayesian full-field response reconstruction approach is proposed for arbitrary distributed loads, enhancing both online and offline methods. In online KF methods, a set of equivalent force vectors derived from the structural dynamic property is used to reduce the dimension of unknown loads, and a reduced system model is obtained with sufficient controllability on the origin system model. The joint input-state estimation filter is utilized to simultaneously identify equivalent loads and full-field responses. In the offline MAP method, a load prior distribution considering spatial correlation is introduced. The most probable load parameters and hyperparameters are iteratively estimated by MAP strategy. Consequently, the identified distributed loads are adopted to reconstruct the full-field structural responses. The improved online and offline methods both do not require prior information of load positions or distribution. Finally, the proposed methods are validated using the structural response data of Qingzhou Bridge to wind and traffic loads.