Organic neuromorphic device networks can accelerate neural network algorithms and directly integrate with microfluidic systems or living tissues. Proposed devices based on the bio-compatible conductive polymer PEDOT:PSS have shown high switching speeds and low energy demand. However, as electrochemical systems, they are prone to self-discharge through parasitic electrochemical reactions. Therefore, the network's synapses forget their trained conductance states over time. This work integrates single-device high-resolution charge transport models to simulate entire neuromorphic device networks and analyze the impact of self-discharge on network performance. Simulation of a single-layer nine-pixel image classification network commonly used in experimental demonstrations reveals no significant impact of self-discharge on training efficiency. And, even though the network's weights drift significantly during self-discharge, its predictions remain 100% accurate for over ten hours. On the other hand, a multi-layer network for the approximation of the circle function is shown to degrade significantly over twenty minutes with a final mean-squared-error loss of 0.4. We propose to counter the effect by periodically reminding the network based on a map between a synapse's current state, the time since the last reminder, and the weight drift. We show that this method with a map obtained through validated simulations can reduce the effective loss to below 0.1 even with worst-case assumptions. Finally, while the training of this network is affected by self-discharge, a good classification is still obtained. Electrochemical organic neuromorphic devices have not been integrated into larger device networks. This work predicts their behavior under nonideal conditions, mitigates the worst-case effects of parasitic self-discharge, and opens the path toward implementing fast and efficient neural networks on organic neuromorphic hardware.