This paper investigates the use of the reconfigurable dual-functional surface to guarantee the full-space secure transmission in non-orthogonal multiple access (NOMA) networks. In the presence of eavesdroppers, the downlink communication from the base station to the legitimate users is safeguarded by the simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS), where three practical operating protocols, namely energy splitting (ES), mode selection (MS), and time splitting (TS), are studied. The joint optimization of power allocation, active and passive beamforming is investigated to maximize the secrecy energy efficiency (SEE), taking into account the imperfect channel state information (CSI) of all channels. For ES, by approximating the semi-infinite constraints with the S-procedure and general sign-definiteness, the problem is solved by an alternating optimization framework. Besides, the proposed algorithm is extended to the MS protocol by solving a mixed-integer non-convex problem. While for TS, a two-layer iterative method is proposed. Simulation results show that: 1) The proposed STAR-RIS assisted NOMA networks are able to provide up to 33.6\% higher SEE than conventional RIS counterparts; 2) TS and ES protocols are generally preferable for low and high power domain, respectively; 3) The accuracy of CSI estimation and the bit resolution power consumption are crucial to reap the SEE benefits offered by STAR-RIS.