In your situation, I would just be happy you have a lovely new fence, that you did not have to pay for. Our stingy scrounging neighbours wouldn't contribute to replacing the rotten falling apart old fence. When DH approached them, he didn't even ask for money, as they pretty much said they agree "as long as you're paying for it." We knew they would never contribute, so did not ask. It honestly wasn't worth the hassle anyway, as they would have opted for the cheapest fence they could find, if by some miricle they had have contributed.
They even approached when the date changed asking if it was still going ahead, as were scared we had changed our mind. We updated them all of the way, apart from the delay, as were waiting for the new date. They hadn't bothered to make any preparation for the job anyway.
They didn't move any of they crap they had piled along the entire boundary, sandwiched against the old fence. We even kindly had their side of the alley replaced too. It is an excellent quality solid fence, and they haven't even said it looks nice, or thank you. Some people are just ignorant, and CF's.
I want them to knock and ask to paint their side, so definately knock on your neighbour. I've paid enough money out, so if they think we are paying for paint for their side they can sod off. We already had to do the entire garden, and have gone through 6 tins of decent paint!! If they actually do anything it'll be cheap rubbish paint any way. 🙄