Hybrid precoding is a cost-efficient technique for millimeter wave (mmWave) massive multiple-input multiple-output (MIMO) communications. This paper proposes a deep learning approach by using a distributed neural network for hybrid analog-and-digital precoding design with limited feedback. The proposed distributed neural precoding network, called DNet, is committed to achieving two objectives. First, the DNet realizes channel state information (CSI) compression with a distributed architecture of neural networks, which enables practical deployment on multiple users. Specifically, this neural network is composed of multiple independent sub-networks with the same structure and parameters, which reduces both the number of training parameters and network complexity. Secondly, DNet learns the calculation of hybrid precoding from reconstructed CSI from limited feedback. Different from existing black-box neural network design, the DNet is specifically designed according to the data form of the matrix calculation of hybrid precoding. Simulation results show that the proposed DNet significantly improves the performance up to nearly 50% compared to traditional limited feedback precoding methods under the tests with various CSI compression ratios.