If you want to build a fence on your property line you might want to think again, especially if you are having trouble with your neighbors. It would be best to build at least a foot away from the line because chances are there could be a bit of a mistake in the deed and you could actually end up building it on the neighbor's property. It is completely legal to build it right on the line if you choose to.
By law you can build a fence on your property line but it may not be the best idea. Often times the property lines aren't accurately marked which could cause the builder of the fence to accidentally place the fence on the property of the other homeowner which could cause numerous problems.
Property lines define the space or area by which you have authority. Fences within your own property will not bother you for any property trespassing, public or private. You just have to check for zoning officials at local offices near you for permits.