Through offloading computing tasks to visible satellites for execution, the satellite edge computing architecture effectively issues the high-delay problem in remote grids (e.g., mountain and desert) when tasks are offloaded to the urban terrestrial cloud (TC). However, existing works are usually limited to offloading tasks in pure satellite networks and make offloading decisions based on the predefined models. Additionally, runtime consumption for offloading decisions is rather high. Furthermore, privacy information may be maliciously sniffed since computing tasks are transmitted via vulnerable satellite networks. In this paper, we study the task-offloading problem in satellite–terrestrial edge computing networks, where tasks can be executed by satellite or urban TC. A privacy leakage scenario is described, and we consider preserving privacy by sending extra random dummy tasks to confuse adversaries. Then, the offloading cost with privacy protection consideration is modeled, and the offloading decision that minimizes the offloading cost is formulated as a mixed-integer programming (MIP) problem. To speed up solving the MIP problem, we propose a deep reinforcement learning-based task-offloading (DRTO) algorithm. In this case, offloading location and bandwidth allocation only depend on the current channel states. Simulation results show that the offloading overhead is reduced by 17.5% and 23.6% compared with pure TC computing and pure SatEC computing, while the runtime consumption of DRTO is reduced by at least 42.6%. The dummy tasks are exhibited to effectively mitigate privacy leakage during offloading.