This work investigates the performance of a downlink non-orthogonal multiple access (NOMA) based coordinated low-earth orbit (LEO) satellite system. Two LEO satellites are assumed to coordinate their transmissions to serve three users simultaneously using NOMA protocol, where only one user lies in the intersection of the footprints of the two satellites. We investigate the reliability of the proposed architecture, which is expressed in terms of the outage probability (OP). We derive closedform expressions of the users' OPs taking into considerations different realistic losses effects on the link budget including receiver antenna gain, satellite antenna gain, antennas pointing errors, shadowing, smallscale fading, and large-scale free space path loss. The channels between satellites and the three users are assumed to follow shadowed-Rician (SR) fading. The mathematical analysis is verified by the extensive representative Monto-Carlo simulations. Finally, we demonstrate the impact of the system parameters on the considered system as well as the superiority of the NOMA scheme over conventional OMA system.