If people have to live space on their private land for sidewalks, then when the city does build the sidewalks they are building privately-owned sidewalks.
The difference between public roads/sidewalks and private ones is called "public right of way". Google it. Chances are if you own property along a public street, you own the land to the middle of the street but for all practical purposes, it's controlled by the local government. You have to allow people to travel across your land. This includes sidewalks within the right of way area and they usually are within the right of way.
