my neighbours have put up a 2 metre fence which is not a problem. However, it is wooden and we would like to paint our side of it and then grow climbing plants against it. What is our situation if we paint something that in theory doesn't belong to us. As a matter of interest our neighbour did not serve a party wall notice on us and when we mentioned this to him he said he didn't have to but whilst this is not an issue we are hoping to put a 2 metre fence up in our front garden and wondered whether we are correct in our assumption that we have to serve a notice on our neighbours and what happens if, like our other neighbours, we don't give this notice.
Regarding painting your neighbours fence, you need their permission to paint it because as you say, it doesn't belong to you. They also don't need to give you notice provided it is on their land.
You shouldn't really grow supporting plants up it either without asking permission.
Everything said about the party wall notice is correct - it is not necessary in this instance.
You may not be able to put a 2m fence in your front garden -look at the covenants on your deeds. Very often 1m is the maximum height for front gardens.