They're pretty much communal when I visited Tokyo/Kyoto. Pretty much always left the building with a different umbrella than I entered with. Just about everyone uses the same $5-10 black/gray/white/clear umbrella. They all go into the umbrella bucket when you enter a building. Everyone just grabs one on their way out that's close enough to the same as the one the dropped in. Sometimes you leave a building and it's not raining anymore and people forget their umbrella behind. People even just leave them behind before getting on the train and take one from the stop where they get off.