In each session, the last several thousand words (from the user and AI) are kept in a context buffer to be used as additional inputs for the neural network. But I don't think ChatGPT lets you choose the AI's responses for that buffer, so you can't really "train" it in any sense of the word. If you want that functionality, use LLaMa.

It will eventually incorporate user inputs in the model. So yes it won't learn in real time from other users, but at some point those inputs will be fed back into itself.