The Segway is designed to integrate into pedestrian traffic with minimal disruption or impact. It's actually hard to hurt a pedestrian when you're riding a Segway because your natural tendency is to shift your balance to avoid any accident.
Many US States have regulations that explicitly enable the use of Segways on the sidewalk. Click to see the US regulatory situation.
If there is no sidewalk, I ride on the street closest to where the sidewalk should be. Basically, where a bike would ride, in the same direction as the flow of traffic. This may be confusing at first, since a pedestrian is supposed to walk against the flow of traffic if they are on the street. But in my opinion 1) it is safer to ride with traffic than against it.
Throughout the world, Segways may be operated on private property with the permission of the property owner.
In practice, this is a conundrum. Do you ask the owner before you enter a building downtown? What about the shopping mall? Do you assume it's OK unless told otherwise? Wouldn't it be nice to know if you could glide around the mall before you got there? Who do you call? Is there a list somewhere?
If there is such a list, I haven't found it. So here's an attempt to build one. Please add to it by hitting the Edit buttons you see on the page. Add your city if it isn't here yet. Then add useful information on each city page.
Right now the list is alphabetical by city. See Honolulu, HI for a sample page.