... We'll see if we can get our Newly-Resident Attorney to give you some general advice. Just to be clear, it sounds like you're still willing to have a fence and have it on YOUR property. And you want to, basically, know 'whose' fence is it and 'who' can take what down.
The way I see it, you can take down what's on YOUR side of the property line and leave what's on your neighbor's side of the property line alone. Then, you can do what you want. (I'm saying worst-come-to-worst ...) If you do this, technically, you can have a fence and SHE can put one up on her side of the line too. Yes, silly it would look to have 2 fences. No matter what, you can put up a fence on your side of the line and so can she, plus, she can put up a privacy fence, and boom, you spent money you didn't have to.