Nothing in my post contradicts anything above (which is all perfectly correct) but I'll point out that there's a difference between owning something and being responsible for its upkeep.
For example, there's a covenant in the deeds of my house which (bizarrely) makes me responsible for cutting the grass on a small piece of land round the corner from me. I don't own it, I can't see it from my house, it's not even on the same road as my house and it's actually been absorbed into the garden of a house three doors away from me, but I'm still officially responsible for maintaining it!
Conversely, I do own the fence separating my house from that of my neighbour but (since there's no covenant in the deeds) nobody has any responsibility for maintaining it. If it falls down I can tell my neighbour that he can either put up with having no fence between our properties or erect his own.
As others have stated, there's a convention that you own the fence with the posts on your side. That simply comes about because that's the way that most people would erect a new fence fronting onto the street. (i.e. with the 'plain' side facing outwards, and the posts on the inside). If that fence is then continued around the house in the same way, you'll end up owning the boundary fence which has the posts on your side.
However, as has also been indicated, that's simply a convention (and not a rule or law).
Further, as I've already pointed out, the person who owns a fence has no obligation to maintain it (unless a relevant covenant is in place).
Confused? Yes, you and a few million other people likewise!
Chris