The following is a direct copy-&-paste from the answer I posted on Tuesday to a similar qurestion (and I stand by it now):
Here we go again! (This question has come up countless times over the years!).
There is NO fixed 'left/right' rule about who owns a fence.
However if you think about how you'd erect a fence at the front of your property, it would almost certainly be with the posts on the inside of the fence and the boards on the outside. Similarly, people usually put up fences between their property and that of a neighbour with the posts on their own side and the boards on their neighbour's side. So it's most commonly 'the person with the posts' who owns a fence (but it's still NOT a rule!).
It also has to be remembered that owning a fence and being responsible for its upkeep aren't the same thing. Unless there's, say, a covenant in the house deeds which requires someone to keep their fence in good order, there's no obligation upon them to fix or replace it if it gets damaged by the wind. They can simply leave it as it is or remove it altogether (leaving their neighbour to erect their own fence if they want to see one in place).