My response to the question of who pays for the new fence usually invokes Robert Frost and his line “Good Fences Make for Good Neighbors”. Robert Frost was talking mostly about neighbors coming together to work on a common project to repair relationships. This same train of thought applies when it comes to sharing the cost of replacing or repairing a fence shared by two neighbors. This should ideally start before the fence needs repairs. A good relationship with your neighbor makes it easier to have the “we should get a new fence” conversation.
If the fence is deemed to be on your property on the property survey, then you must know your neighbor has nothing requiring them to participate in the financials of the project. You cannot compel your neighbor to fix the fence. However, if both neighbors use the fence and they can come to terms it is usually a good thing to talk with the neighbor on the other side of that fence to share costs of repairs or replacement.
I have seen instances where neighbors split the cost evenly. Sometimes one neighbor is willing to purchase the supplies and the other is willing to perform the labor. Maybe it is easier to hire a fencing company and split the cost evenly. Be sure to discuss which side gets the “ugly” side with the pickets showing or the “good” side which is all planks.
Once you have come to terms with a fence design be sure to consult with your neighborhood HOA or neighborhood architectural control committee to make sure your plans are in line with neighborhood covenants. If you plan to do any work yourself be sure to contact 811 and “call before you dig” to prevent accidentally running into any underground utilities.