We analyze a one-dimensional PDE-ODE system representing the diffusion of signaling molecules between two cells coupled by a stochastically gated gap junction. We assume that signaling molecules diffuse within the cytoplasm of each cell and then either bind to some active region of the cell’s membrane (treated as a well-mixed compartment) or pass through the gap junction to the interior of the other cell. We treat the gap junction as a randomly fluctuating gate that switches between an open and a closed state according to a two-state Markov process. This means that the resulting PDE-ODE is stochastic due to the presence of a randomly switching boundary in the interior of the domain. It is assumed that each membrane compartment acts as a conditional oscillator, that is, it sits below a supercritical Hopf bifurcation. In the ungated case (gap junction always open), the system supports diffusion-induced oscillations, in which the concentration of signaling molecules within the two compartments is either in-phase or anti-phase. The presence of a reflection symmetry (for identical cells) means that the stochastic gate only affects the existence of anti-phase oscillations. In particular, there exist parameter choices where the gated system supports oscillations, but the ungated system does not, and vice versa. The existence of oscillations is investigated by solving a spectral problem obtained by averaging over realizations of the stochastic gate.