Here's the top of my flask and part of the view from the top of Ben Lomond. You can just make out the pipeline for the hydro above the flask. There are two arrows on the flask, indicating two openings. As you pour coffee out through one side, air is able to go in the other side to take its place. If the air isn't able to get in, the coffee won't come out. So in a sense it is air pressure pushing the coffee out of the flask.
