Saliency prediction has been extensively studied in RGB images and videos as a computational model of human visual attention. In contrast, predicting saliency from event-based data remains largely unexplored, despite the biological inspiration and favorable sensing properties of event cameras. Two obstacles have held this direction back: the absence of large-scale event saliency datasets, and the lack of a strong baseline. In this paper, we introduce SEST (Swin Event-based Saliency Transformer), a transformer-based model for saliency prediction from event data, bridging the data scarcity barrier through event-native pretraining and synthetic supervision. SEST leverages a self-supervised pretrained event-based Swin Transformer backbone combined with a lightweight CNN decoder to produce dynamic saliency maps. To address the scarcity of annotated event-based saliency data, we introduce two new benchmark datasets, N-DHF1K and N-UCF Sports, generated from large-scale RGB saliency benchmarks. Experimental results show that SEST clearly outperforms existing event-based saliency methods and narrows the performance gap with state-of-the-art RGB models. Zero-shot evaluation on a real event camera dataset further demonstrates that our model trained on synthetic data remains transferable on real event streams. To the best of our knowledge, this work is the first to apply deep learning to event-based saliency prediction, opening a new research direction at the intersection of event-based vision and neuromorphic visual attention.