There is a couple issues here and you may need to hire an attorney to threaten suit. One issue is the land. It is half on yours and half on hers. You'd be trespassing if you took the fence down on her side.
The other issue is who put the fence up and owns it. If you or the prior owner of your home put it up, then you have ownership of all and possession of the part of the fence on your side. Your neighbor may have a possessory interest and a claim for adverse possession by way of the fence put on her side. Therefore you still can't take the fence down on that side without risking damage to the part on her side and a claim for damages.
If she or a prior owner on that side put it up and they are claiming ownership, then the shoe is on the other foot. They are trespassing on your side and you may have a claim for tresspass and/or adverse possession also. Sometimes neighbors share the cost of the fence and both are owners as tenants in common.
Either way, a written agreement between you two is the way to go. It may take an attorney to get your neighbor moving in that direction by threating a quiet title/trespass/adverse possession action for the fence. Some states have fence laws where the county commissioners review the fences.