Summary: To provide a solid analytic foundation for the module approach to conditional risk measures, our purpose is to establish a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the \((\varepsilon,\lambda)\)-topology and the locally \(L^0\)-convex topology). This paper is focused on the part of separation and Fenchel-Moreau duality in random locally convex modules. The key point of this paper is to give the precise relation between random conjugate spaces of a random locally convex module under the two kinds of topologies, which enables us to not only give a thorough treatment of separation between a point and a closed \(L^0\)-convex subset but also establish the complete Fenchel-Moreau duality theorems in random locally convex modules under the two kinds of topologies.