Agree with all that's been said about the general principles of sorting out what's whose ..... but that said, when our left-hand fence at the back came down last year, it turned out to be our responsibility, so we went ahead and replaced the two panels and post affected - and paid for them. NDN didn't offer to contribute, but he'd only recently moved in and I think he was out of work at the time, so we didn't mind.
The right hand fence (back and front) is not ours, but the responsibility of the Housing Association who are landlords for the first house in the next road. Several years ago, they refused to allow us to put trellis-work on top of it at the front, at our expense (which we wanted to do to stop miscreants climbing over it from their road into our road) saying it would be unsightly. Lucky we asked them, I guess!
So we planted fast-growing red robinia bushes just inside our border instead which are now higher than the fence ....