The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. We first propose a discretization of this system, where the displacements are approximated by standard finite elements and the action by a local postprocessing which admits an equivalent mixed reformulation. We perform the a posteriori analysis of this discretization and prove optimal error estimates. Numerical experiments confirm the efficiency of the error indicators.