Existing automatic sleep stage detection methods predominantly use convolutional neural network classifiers (CNNs) trained on features extracted from single-modality signals such as electroencephalograms (EEG). On the other hand, multimodal approaches propose very complexly stacked network structures with multiple CNN branches merged by a fully connected layer. It leads to very high computational and data requirements. This study proposes replacing a stacked network with a distributed neural network system for multimodal sleep stage detection. It has relatively low computational and training data requirements while providing highly competitive results. The proposed multimodal classification and decision-making system (MM-DMS) method applies a fully connected shallow neural network, arbitrating between classification outcomes given by an assembly of independent convolutional neural networks (CNNs), each using a different single-modality signal. Experiments conducted on the CAP Sleep Database data, including the EEG-, ECG-, and EMG modalities representing six stages of sleep, show that the MM-DMS significantly outperforms each single-modality CNN. The fully-connected shallow network arbitration included in the MM-DMS outperforms the traditional majority voting-, average probability-, and maximum probability decision-making methods.