A novel simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RISs) aided downlink non-orthogonal multiple access (NOMA) communication framework is proposed. Two STAR-RIS protocols are investigated, namely the energy splitting (ES) and the mode switching (MS). However, since the STAR-RIS has a massive number of reconfigurable elements, the passive beamforming problem has enormous action dimensions and extremely high complexity, resulting in an increased training time and performance degradation for the artificial intelligent agent. To resolve this predicament, a partitioning approach is proposed to divide the STAR-RIS into several tiles. A distributed learning approach is conceived for the partitioning and the corresponding tile-based passive beamforming of each STAR-RIS, as well as the power allocation for users to maximize the average throughput achieved by multiple base stations. In particular, the deep reinforcement learning (DRL) agent is employed at each BS, and an access-free federated learning (AFFL) model, which gathers the features of federated learning (FL) and transfer learning (TL) is proposed to accelerate or even exempt the training process for agents. Simulation results indicate that 1) the ES protocol is preferred for being employed in the NOMA networks compared with the MS protocol; 2) the tile-based passive beamforming approach outperforms benchmarks while the STAR-RIS has a large size; 3) the proposed AFFL framework effectively reduces or exempts the training overhead.