Prompt and effective corrective actions in response to unexpected contingencies are crucial for improving power system resilience and preventing cascading blackouts. The optimal load shedding (OLS) accounting for network limits has the potential to address the diverse system-wide impacts of contingency scenarios as compared to traditional local schemes. However, due to the fast cascading propagation of initial contingencies, real-time OLS solutions are challenging to attain in large systems with high computation and communication needs. In this paper, we propose a decentralized design that leverages offline training of a neural network (NN) model for individual load centers to autonomously construct the OLS solutions from locally available measurements. Our learning-for-OLS approach can greatly reduce the computation and communication needs during online emergency responses, thus preventing the cascading propagation of contingencies for enhanced power grid resilience. Numerical studies on both the IEEE 118-bus system and a synthetic Texas 2000-bus system have demonstrated the efficiency and effectiveness of our scalable OLS learning design for timely power system emergency operations.