As carbon emissions reduction is becoming increasingly important for sustainable development and carbon neutrality targets, the concept of carbon emission market has been recently proposed in order to essentially manage carbon emission on the demand side by allowing electricity consumers to purchase or sell carbon emission quotas. Hence, real-time demand-side carbon emission monitoring (DCEM), indicating the amount of carbon emission that each electricity consumer should be responsible for, becomes a necessity for the operation of the carbon emission market. However, the real-time DCEM cannot be achieved by carbon emission flow (CEF) analysis due to the unavailability of real-time power demand data. In this connection, this paper proposes a two-stage real-time DCEM method based on the graph neural network (GNN). In the first stage, power system operating scenario data, including the power generation capacity and power demand data, are collected to calculate carbon emission patterns through CEF analysis. In the second stage, a data-driven GNN-based model is designed to learn from historical daily carbon emission patterns and then realize accurate real-time DCEM with real-time available generation-side measurements only. Case studies on the 118-bus power system operated with day-ahead planning considering carbon emission are performed to demonstrate the accuracy and effectiveness of the proposed method.