once a neighbor puts up a fence it becomes a “common” fence
Actually, I’m not sure I would agree with that. There are certain situations where that can be the case, certainly in New Orleans.
Where a property owner puts up a fence on his property, it is his fence sitting on his property. A typical subdivision fence does not straddle the property line with 2 inches of the post on one side and 2 inches on the other. Assuming that is the case, then permission would be necessary to alter the fence.