Automatic sleep staging based on electroencephalography (EEG) and electromyography (EMG) signals is an important aspect of sleep-related research. Current sleep staging methods suffer from two major drawbacks. First, there are limited information interactions between modalities in the existing methods. Second, current methods do not develop unified models that can handle different sources of input. To address these issues, we propose a novel sleep stage scoring model sDREAMER, which emphasizes cross-modality interaction and per-channel performance. Specifically, we develop a mixture-of-modality-expert (MoME) model with three pathways for EEG, EMG, and mixed signals with partially shared weights. We further propose a self-distillation training scheme for further information interaction across modalities. Our model is trained with multi-channel inputs and can make classifications on either single-channel or multi-channel inputs. Experiments demonstrate that our model outperforms the existing transformer-based sleep scoring methods for multi-channel inference. For single-channel inference, our model also outperforms the transformer-based models trained with single-channel signals.