The combination of simultaneously transmitting and reflecting-reconfigurable intelligent surface (STAR-RIS) and non-orthogonal multiple (NOMA) brings the necessary full-space degrees of freedom and spatial multiplexing gains for the Internet of Things (IoT) networks. The inherent network heterogeneity and sharing of wireless channels may however increase the exposure of the information interactions to the third party. To address this issue, we propose a covert communication scheme in STAR-RIS assisted NOMA networks over Nakagami-m fading channels, where both downlink and uplink IoT scenarios are considered. Under the NOMA protocol with imperfect successive interference cancellation (SIC), an IoT access point interacts with two IoT users aided by a STAR-RIS without being detected by two wardens. In this scenario, the two IoT users are located on both sides of the STAR-RIS which adopts coherent phase shifting and operates according to the mode switching protocol. To evaluate the wardens’ detection performance, the Kullback-Leibler (KL) divergence is used. Furthermore, the cascaded channel gains of IoT users and wardens are respectively characterized as Gamma and complex Gaussian random variables. The closed-form expressions of the expectations of KL divergence and the interruption probabilities for downlink and uplink are derived. To further improve the performance, we formulate the effective covert rate maximization as the joint optimization problems of the transmit power and power allocation coefficient for downlink and uplink, subject to the constraints for covertness, reliability and power budget, which are respectively resolved analytically. Extensive simulation results indicate that the proposed scheme improves covertness compared with the benchmarks.