This paper studies the performance for the proactive eavesdropping-based legitimate surveillance for the integrated satellite-terrestrial relay networks in the presence of several monitors. To investigate the proactive eavesdropping performance, we propose three different presentative proactive eavesdropping cases/modes. In these cases, the monitors can chose its roles, i.e., the eavesdroppers or jammers. On this foundation, three cases are considered in this paper, namely, Case I: Eavesdropping then Jamming, Case II: Jamming then Eavesdropping, Case III: Always Eavesdropping. Especially, the exact expressions for the eavesdropping non-outage probability for the considered three modes are obtained. Besides, the asymptotic expressions for the eavesdropping non-outage probability are further derived based on the three proposed cases. In addition, the colluding proactive eavesdropping algorithm is utilized in the considered proactive network, which means all the monitors works together to steal the information or jam the users. At last, some Monte Carlo results are given to confirm the validity of the theoretical results.