To address the problem of separating images into a structure and a texture component, we introduce a joint structure-texture model. Instead of considering two separate regularizations for each component, we consider a joint structure-texture model regularization function that takes both components as inputs. This allows for the regularization to take into account the shared information between the two components. We present evidence that shows a performance gain compared to separate regularization models. To implement the joint regularization, we adapt the plug and play framework to our setting, using deep neural networks. We train the corresponding deep prior on a randomly generated synthetic dataset of examples of this model. In the context of image decomposition, we show that while trained on synthetic datasets, our plug and play method generalizes well to natural images. Furthermore, we show that this framework permits to leverage the structure-texture decompositions to solve inverse imaging problems such as inpainting.