This question has come up countless times on AB. The simple answer is that there's no definitive answer.
However there's a convention (NOT a rule!) that house-owners construct their fences with the posts on 'their' side, so that's a good starting point.
It's also worth remembering that 'ownership' and 'responsibility' aren't synonymous. You might, for example, own a fence that's falling down. Unless there's a relevant covenant in the deeds to your property (requiring you to maintain the fence) there's nothing that obliges you to fix it. If your neighbour complains, you can simply take your fence down (or invite him to do so) and say "If you want a fence, then you put one up. I can't be bothered with it".