We investigate a closed-loop supply chain (CLSC) where the retailer and the third-party recycler compete against each other to collect waste electrical and electronic equipment (WEEE) given that collection effort is their private information. Using the principle-agent theory, we develop a CLSC model with dual collection channel without the government’s reward–penalty mechanism (RPM). An information screening contract is designed for the manufacturer to attain real information on collection effort levels; meanwhile, the optimal decision-making results of other decision variables are derived. Next, we take RPM into account to further examine the efficacy of the government’s guidance mechanism in improving collection rate and profits of CLSC members. Our results indicate that (i) the collection competition reduces the total collection quantity and the expected profits of all the CLSC members without RPM; (ii) all CLSC members’ expected profits are improved if both two collection agents select a high collection effort level without and with RPM; (iii) RPM increases buyback price, collection price, collection quantity, and franchise fee but decreases wholesale price and retail price; with the reward–penalty intensity increasing, the manufacturer’s expected profit first decreases and then increases, while the expected profits of H-type retailer and H-type third-party recycler continue to increase. We find that RPM may ultimately stimulate the collection agents to collect more WEEEs, while the intense collection competition reduces the profits of CLSC members.