The front of my house was covered in Virginia creeper until one of the spring gales brought two thirds of it down. I have contemplated having the rest down, but the root is in my garden and the creeper has spread onto my neighbour's house so theirs would die. Over the years, my neighbours have actively encouraged the growth. Who would be responsible for taking down the dead Virginia creeper and disposing of it and who would bear the cost?
It can be very difficult to get rid of Virginia Creeper. It is considered an invasive plant.
Here is a link that may help with suggestions on how to get rid of it.
As to who will pay for any costs perhaps your neighbor can split any cost incurred.