I think you are blending two different things.
Everyonje, to a degree, practices 習字 which means "studying characters", and it is done at grammar schools, mainly. That is not what the man on the video is doing, as he is writing calligraphy (書道, or 書法 in chinese). Shuuji (習字) is not art, its like western calligraphy - a craft - teaching you how to write pretty characters. Shodou (書道) is pure art, and even though there are rules to it, it has nothing to do with pretty charatcers.
In Japan - I have not seen anyone doing that (writing on a street). In Japan calligraphy is practiced at home, or during special studying meetings (勉強会). There are occasionally organised outdoor events. Like this:
Koji Kakinuma, Japanese Calligrapher/Artist
There is one guy in Ueno park writing kanji of people names, but despite an original handwriting (書風), he is lacking basics.
The guy on the video, on the other hand, is wicked (in fact both of them are, the Chinese bloke and the one i linked to).
If you want to read more about calligraphy, click on the links in my signature.