Does moving a deal to a new stage notify the other side automatically?

Yes. When you move a deal between stages on the board view, it can automatically send a notification to keep the other side in the loop, so you do not have to send a separate update. If you would rather move a deal quietly, you can do that too and update people yourself when you are ready. It is easy to miss at first because the board looks like a simple internal pipeline, but those stage changes can do the notifying for you.