This paper proposes a distributed learning-based framework to tackle the sum ergodic rate maximization problem in cell-free massive multiple-input multiple-output (MIMO) systems by utilizing the graph neural network (GNN). Different from centralized schemes, which gather all the channel state information (CSI) at the central processing unit (CPU) for calculating the resource allocation, the local resource of access points (APs) is exploited in the proposed distributed GNN-based framework to allocate transmit powers. Specifically, APs can use a unique GNN model to allocate their power based on the local CSI. The GNN model is trained at the CPU using the local CSI of one AP, with partially exchanged information from other APs to calculate the loss function to reflect system characteristics, capturing comprehensive network information while avoiding computation burden. Numerical results show that the proposed distributed learning-based approach achieves a sum ergodic rate close to that of centralized learning while outperforming the model-based optimization.