In this paper, the legitimate surveillance of a suspicious mobile edge computing (MEC) network consisting of a suspicious edge server (SES) and multiple suspicious users (SUs), in the presence of a full-duplex monitor is studied. Each SU has a computation task to complete within a time deadline and can completely or partially offload the task to SES, while the monitor can either jam or assist the suspicious communications during task uploading and result downloading. With the heterogeneous offloading model adopted by the SUs, the problem of optimizing the monitor mode and transmit power to maximize the average ratio of successfully eavesdropped tasks, subject to the monitor transmit power constraint and the task completion time deadline constraint is investigated. The problem is solved via exploring the particular problem structure and adopting the sum-of-ratios optimization. Simulation results show that the proposed algorithm significantly outperforms the benchmark algorithms, especially for the SUs with partial offloading. It is also shown that the proposed algorithm is of low-complexity and achieves almost the same performance as the high-complexity optimal algorithm. Besides, compared to the SUs with binary offloading, the eavesdropping performance for the SUs with partial offloading is shown to be much better.