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 transmission overhead, data privacy, and computation load. 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 (SSCNet) with the lightweight structure is designed for the edge device to compress the collected raw data into a compact semantic message that is then sent to the edge server via the wireless channel. On the edge server side, a modulation classification neural network (MCNet) combining bidirectional long short-term memory (Bi-LSTM) and multi-head attention layers is elaborated to determine the modulation type from the noisy semantic message. By leveraging the computation resources of both the edge device and the edge server, high transmission overhead and risks of data privacy leakage are avoided. The simulation results verify the effectiveness of the proposed C-AMC framework, significantly reducing the model size and computational complexity.