This article investigates the detection and identification of physical faults in devices and false-data-injection attacks in communication networks for multi-agent systems with event-triggered transmission mechanisms and subject to external periodic disturbances. First, a new detection and identification scheme, including a local disturbance decoupling (LDD) observer and a distributed disturbance decoupling (DDD) observer, is proposed. Then, based on zero-assignment and the rank-deficiency of the transfer function matrix at zeros, a co-design method for the LDD observer and DDD observer is proposed, which enables the decoupling of periodic disturbances from the residuals for detection and identification. This new scheme no longer requires the transmission of control signals from the node being monitored or the exchange of information between its neighbors, significantly reducing the communication overhead and enhancing the system's security. Finally, a simulation based on a multi-two-wheeled trolley system is used to verify the effectiveness of the proposed method.