I live on 2.5 acres. We have recentlymoved from a gated community where our only had access to back yard. Now that we've moved to this property our dog is curious, she has never been around other dogs freely. The first month we moved to this property she engaged in a dog fight with our neighbors dog. Since then our neighbor has insisted we keep our dog leashed up and muzzled(unfair) We have done everything to make peace with this neighbor she feels she has a say so as to what we do on our property. I totally understand her panic of being attacked again but come on its been a year our dog stay in the area where we see her and comes when we call. Now the twist to this is that our property extends down pass her house, and when we walk down the hill to our mail box her dog comes out barking at me and my kids she also allows her dog to run freely on lower property in which my down can never go down in because its close to her house. Can she inforce rules on my property regarding my dog, even if after a year the dog has not coused any neighborhood issues. She has also passed the word to surrounding neighbors we have a vicious dog in which we don't know how to maintain. What right do I have?

You can keep the peace and follow rules no matter what the property lines are. Why would you want to make bad feelings over dogs. I don't here. I have invisible fencing, my dog can't go off her property. We have an on lease rule in our town which we all maintain. It is just understanding to maintain boundaries. Do you want to have bad relations with your neighbours; have a meeting between you all and figure it out.

Either that, or hire a lawyer and spend money to clear it up.

Simple

Tick

You have to follow local ordinances regarding animals. You need to find out what they are.
Your town or county may or may not have rules about fences or leads. It is easy enough to string a long cable from house to a tree or post and have a traveling lead attached to your dog's collar. Then, if her dog wanders onto your property and is attacked, she has a lot less of a case. But the law in most states says that injury can be the fault of the owner of the land it happened on, if the owner is deemed negligent (hole in ground, attacking dog, pail of chemicals, anything). This sort of thing sometimes ends up in court for a judge to decide (since pets are 'property,' small claims). Sometimes the police will come and strongly suggest a resolution, or they will call the animal warden, who can decide, and who can take an animal away.
In short, you don't have to do what she says, but you need to know local law and the consequences.
And what about a small child wandering onto your land? Do you know your dog won't attack?

In addition, it seems to me that a dog owner who leaves his or her dog unrestrained doesn't care that much about the dog. Any dog which is not kept in a fenced yard, tied, or on a leash runs the risk that the dog will go off-property and get run over.

We have been having a lot of smaller, running loose dogs disappearing in the Atlanta area. Can you spell Coyote!

Coyote, yes I can spell it, I saw one off 285 running down in the west area about a month ago. ( that is atlanta area)
You get down south of woodstock you will even have some other animal issues.

But to address the question, your neighbor can not enforce anything, they can of course call and report you to animal control, or the police. If your dog comes onto their property, or if it goes into the street and attacks their dog, she can sue you for the damages.

Have you considered things like the INVISIBLE fence or even a real fence. So that your dog is protected from running off and going on to others property.

Allowing a dog to run free is contrary to all the dog training I had gone through. You need to control your dog or your dog will control you. I would suspect your dog already does not see you as his alpha since he has already had dog fights. I agree with AK, if you love your dog then control him. Do not allow free range. Joypilv legal angles are all right on point. The laws in your area will be bthe test so find out what they are. If they do allow free range you are still liable for what you dog does to other dogs and people.