We examine the role of interactions when a Bose gas, in a double-well potential with a finite tunneling probability (''Bose-Josephson junction''), is exposed to external noise. We examine the rate of decoherence of a system initially in its ground state with equal probability amplitudes in both sites. The noise may induce two kinds of effects: firstly, random shifts in the relative phase or number difference between the two wells and secondly, loss of atoms from the trap. The effects of induced phase fluctuations are mitigated by atom-atom interactions and tunneling, such that the dephasing rate may be suppressed by half its single-atom value. Random fluctuations may also be induced in the population difference between the well, in which case atom-atom interactions considerably enhance the decoherence rate. A similar scenario is predicted for the case of atom loss, even if the loss rates from the two sites are equal. We find that if the initial state is number-squeezed due to interactions, then the loss process induces population fluctuations that reduce the coherence across the junction. We examine the parameters relevant for these effects in a typical atom chip device, using a simple model of the trapping potential, experimental data, and the theory of magnetic field fluctuations near metallic conductors. These results provide a framework for mapping the dynamical range of barriers engineered for specific applications and sets the stage for more complex circuits (''atomtronics'').