In KY, the standard is that a fence is placed on the line and cost is shared by both sides. Basically, you 'meet face to face in the middle of the property line, and Each person is liable for upkeep of the fence 'to his right'. That is the basic 'legal' definition which is assumed by the courts. However, if a neighbor wants to 'control' the fence, how it is built, materials, etc. He must put it on his own property and care for it himself. This keeps a 'Rich' neighbor from moving in, and requiring a 'poor' neighbor to pay for 1/2 of a 4 board fence.
Personally, I have several miles of fence to put in over the next couple years. I want control of the fence and I want to be able to mow on both sides. (I'm in KY Horse country). I spoke with my neighbor and explained that I would pull the old and care for the new fence. But that I didn't cede my property on the 'other' side of the fence, and that the property line remained as surveyed. He was very happy, because the fence I pulled was honeysuckle grown over 50yo barbed wire which was nothing but a nuisance. The 'line' is lined with old walnut, oak, etc. So, trying to know where it is, isn't difficult. Caring for it is a PITA, so I'm moving it in quite a bit, to make life easier.
I still need to decide 'how far' inside I want to go. But that is simply a decision based on 'mowability' of the other side. Which is what I am pondering at this point.