Where I live I would have to ask, what the hell is a neighbor?
I respect their wishes while on their property and would not carry.
What you might consider is once you have established a friendly relationship reciprocate and invite them over to your place where you are carrying thus, without offense you create an opportunity to start a conversation.
The biggest hurdle we have is that for the vast majority of people in the US their only real reference for people and firearms is the crime they see reported on the nightly news. Thus, they logically begin to always associate guns with criminals.
If we want to bring more people over to our side of the debate we simply need to work to expose as many as possible of them, in the least confrontational manner we can to normal, average people going about their daily lives carrying as responsibly armed Americans.
Someone that lives a half mile or more from me.