Short answer is clearly no you can't paint it without their permission, nor grow anything up it, or attach anything to it.
We have a shared fence with the neighbour, we recently went halves on replacing some damaged panels. It was a simple process of knocking on their door, introducing ourselves and chatting about the fence.
We have lived here a couple of years but due to the layout of the houses they aren't next door to us, they are 20 houses away but the road backs round on itself, so we never actually see them due to the 6ft fence!
It was clear when we saw the fence from their side that the new untreated panels stuck out like a sore thumb and our side (inherited) has some panels treated and some not due to shrubbery that we have now removed.
We asked them what they treated their side with and agreed we would do the same on our side so that any leakage through the panel wouldn't spoil any painting they had done on their side.
It is really easy to talk to the neighbour about it rather than start doing stuff and it starting a dispute. Have a look at Gardenlaw forum and particularly this thread where the police were called over someone painting a fence [[http://www.gardenlaw.co.uk/phpBB2/viewtopic.php?f=6&t=18520 here]