The area you are referring to in most cities/towns/villages is the parkway or easement area. It's technically owned by the city but maintained by the homeowner. I am guessing by "walk your dog" they mean stop and do his business. I would imagine that because the homeowner maintains that area they feel that they would rather you not" walk your dog" there. If they are actually referring to you walking your dog in front of their house, then the sidewalk is a public space and they can not say anything about that. Check with your local ordinance to be sure.
I have always heard that the area that is within 3-6 feet from the curb (Called the devilstrip in my hometown) is the cities property,not any given home owners,although the home owner takes care of that property as their own. That may be why the home owners are telling you that, and feel it is their property,when in fact the city owns it for issues such as public utilities go,and depending on where the property is located, even sewer and other public utility access areas go. So,your best bet is to call the city and ask them for sure!
Good luck and please let us know what you find out. Being a pet owner myself, I am very curious as to their answer. :-)