This brief investigates the regulation problem for a class of networked nonlinear systems with measurement noise, where random data dropouts in both the feedback and forward channels are considered. To actively compensate for the two-channel data dropouts, a data-driven networked compensation control method is proposed, which consists of two aspects: 1) to calculate a control increment based on the measured output error in the controller and 2) to design a data dropout compensation strategy based on the latest control increment available in the actuator. The proposed method merely depends on the input and output data of the controlled plant, without using explicit or implicit information of its mathematical model. Moreover, only one control command needs to be transmitted in the forward channel at each time instant. A sufficient condition is derived to guarantee the closed-loop stability and output error convergence. Both numerical simulations and experimental tests are conducted to demonstrate the effectiveness of the proposed method.