You might think that you'd use "on the street" to describe a person who's walking. You do use "on" when someone is just standing:

I saw her standing on the street.

But when someone is walking or running, they do it "down" the street like in the example at top.

You can also use "up the street" when someone's moving away from you:

If you walk up the street a few blocks, you'll see a florist on your right.