Due to both of jamming and eavesdrop-ping, active eavesdroppers can induce more serious security threats to unmanned aerial vehicle (UAV)-enabled communications. This paper considers a se-cure UAV communication system including both the downlink (DL) and uplink (UL) transmissions, where the confidential information is transmitted between a UAV and a ground node in the presence of an active eavesdropper. We aim to maximize the average se-crecy rates of the DL and UL communications, respec-tively, by jointly optimizing the UAV trajectory and the UAV/ground node's transmit power control over a given flight period. Due to the non-convexity of the formulated problems, it is difficult to obtain globally optimal solutions. However, we propose efficient iter-ative algorithms to obtain high-quality suboptimal so-lutions by applying the block coordinate descent and successive convex optimization methods. Simulation results show that the joint optimization algorithms can effectively improve the secrecy rate performance for both the DL and UL communications, as compared with other baseline schemes. The proposed schemes can be considered as special cases of UAV-assisted non-orthogonal multiple access (NOMA) networks.