This paper investigates the secrecy outage performance for a large-scale downlink system with full-duplex (FD) non-orthogonal multiple access (NOMA) transmission aided by artificial noise (AN). In the system, legitimate users (LUs) are randomly distributed according to homogeneous Poisson point processes whereas a passive eavesdropper (Eve) is certainly located. We introduce a secure cooperative communication scheme, in which nearby NOMA users operating in FD mode act as jammers and generate AN to enhance the physical layer security (PLS) of the legitimate transmission. Closed-form expressions in terms of the secrecy outage probability (SOP) for a user pair are acquired. Following the mathematical results, secrecy diversity order analysis is further carried out for two scenarios, i.e., 1) fixed transmit signal-to-noise ratio (SNR) at the Eve, and 2) transmit SNR at the Eve being proportional to that of LUs. Closed-form expressions of asymptotic SOP for the two scenarios are also obtained. Monte Carlo simulations are performed to certify the veracity of the theoretic analysis and illustrate that the proposed scheme employing both AN and FD-NOMA is superior in the secrecy outage to the NOMA scheme without AN and the orthogonal multiple access scheme with AN. Furthermore, the SOP performance enhancement through proper secrecy rate target selection and power allocation optimization is also evidenced by the simulations.