I know in my area of PA, when we put a fence up, it should be so many feet from the property line.
If it is on the property line, it does not matter who paid for it. The side that is on the owner's property line is theres. The other side that is on the neighbor's property is the neighbors. If they want to paint that side, they can. If they want to hang something on that side, they can. That fence is now on their property.
Now, usually, if you are going to do this, you have talked to the neighbors about it, sometimes, each paying for half the fence.
But, you never know if one neighbor sells their house, what the new neighbors will do. They do not have to abide by the original agreement of the previous owners and the neighbors.
If the original owners of the blackberry bush did not want their neighbors to be picking off of the bush, then they should have planted the bush somewhere where they could control their own bush.
As to the OP, I see no problem with her picking and enjoying the blackberries. If the neighbor has a problem with this, he can trim the branches from his side of the fence.