Ideally you and the neighbor would agree on a plan, and share the cost of the tree removal and fencing. I have had a wooden privacy fence start and continue on either side of a large oak tree, leaving the tree in place and it looks pretty good.
Since you and the neighbor don't get along, what I suggest is to first have a survey done to be sure where the property lines are, and who owns the tree. If the tree is, in fact, straddling the line, then I would consider contacting the local constable or justice of the peace to try to arbitrate.

You could also design a shrub border for privacy and incorporate the tree into the border. I have such a situation with my neighbor (we do get along), and I offered to take care of the bed and the tree, if they would allow me to run my bed edging around the tree onto their property. My shrub border is far more interesting than a board fence and I seriously doubt it cost as much. Just another thought.

Getting a survey to confirm the exact property line is the first step. If it does indeed split the line, then you need to see a lawyer about who would be responsible if it fell and there was damage to either piece of property. Also, this will help you be sure where to erect your fence. Be sure the fence is on your side of the line. Our fence was built before I moved into this house. Our house was the second one on the street and the house next door was built after our fence was erected. The neighbors never complained until they moved and the new folks moved in. The 175 fence was 3 feet over their line on the back side of my property but on the front side of theirs. I had a new survey so I would know exactly where the line is. I moved the fence and it is indisputably on my side of the line. The lady next door wanted it moved 3 feet inside my line so they could park their boat but I refused since she had been so ugly about my fence being 3 feet on her side. Needless to say, she doesn't speak to us.