The simulation of multi-domain, multi-physics mathematical models with uncertain parameters can be quite demanding in terms of algorithm design and computation costs. Our main objective in this paper is to examine a physical interface coupling between two random dissipative systems with uncertain parameters. Due to the complexity and uncertainty inherent in such interface-coupled problems, uncertain diffusion coefficients or friction parameters often arise, leading to considering random systems. We employ Monte Carlo methods to produce independent and identically distributed deterministic heat-heat model samples to address random systems, and adroitly integrate the ensemble idea to facilitate the fast calculation of these samples. To achieve unconditional stability, we introduce the scalar auxiliary variable (SAV) method to overcome the time constraints of the ensemble implicit-explicit algorithm. Furthermore, for a more accurate and stable scheme, the ensemble data-passing algorithm is raised, which is unconditionally stable and convergent without any auxiliary variables. These algorithms employ the same coefficient matrix for multiple linear systems and enable easy parallelization, which can significantly reduce the computational cost. Finally, numerical experiments are conducted to support the theoretical results and showcase the unique features of the proposed algorithms.