It's a double skinned fence (I think that's what it's called). I have my side and neighbour has exactly the same on their side. the fence boards over lap slightly so you can't see through the gaps. The boards on both sides are fitted to one middle line of fence posts. The gap between the top of mine and the top of theirs is big enough for my cat to lie in (this is a standard form of measurement, right?) 
I hope I have put enough information there. I want to put some trellis along the top for added privacy but I suspect it will be tricky to do without having access to neighbour's garden which isn't an option.
Has anyone done this on a similar kind of fence?