In hierarchical cognitive radio networks, edge or cloud servers utilize the data collected by edge devices for modulation classification, which, however, is faced with problems of the computation load, transmission overhead, and data privacy. In this article, an edge learning (EL) based framework jointly mobilizing the edge device and the edge server for intelligent co-inference is proposed to realize the collaborative automatic modulation classification (C-AMC) between them. A spectrum semantic compression neural network is designed for the edge device to compress the collected raw data into a compact semantic embedding that is then sent to the edge server via the wireless channel. On the edge server side, a modulation classification neural network combining the bidirectional long-short term memory and attention structures is elaborated to determine the modulation type from the noisy semantic embedding. The C-AMC framework decently balances the computation resources of both sides while avoiding the high transmission overhead and data privacy leakage. Both the offline and online training procedures of the C-AMC framework are elaborated. The compression strategy of the C-AMC framework is also developed to further facilitate the deployment, especially for the resource-constrained edge device. Simulation results show the superiority of the EL-based C-AMC framework in terms of the classification accuracy, computational complexity, and the data compression rate as well as reveal useful insights paving the practical implementation.