Those distributors usually do not offer tissues to foreigners, by the way. However, in the hustle and bustle of the subway stations, we managed to grab some several times. As for using the tissues: place them on the shelf *before* washing your hands—rummaging in your pocket with wet fingers after washing is uncool.
By the way, in Japan, it is considered extremely rude to blow your nose in public. This is even stricter than in the USA, where you are not supposed to blow your nose at the table and should excuse yourself to go to the restroom instead.