This is an interesting topic... some people think the fence is owned by whoever paid for it, but this isn't always true.
Fence ownership is usually decided by which property the fence is located o,n and this can only be verified by a survey provided by a licensed professional surveyor. If the fence is within your property line then you own 100% of the fence. If, by some chance a fence is directly on the property line, then it would seem that you and your neighbor both own the fence equally.
Please verify any questions of property rights and ownership with an attorney.