You can cut back any growth that's hanging over your boundary but you have to offer it back to them.
if you are on good terms with them why not say that you intend to trim the overhanging bits off and do they want the material or should you just compost it/bin it?

they may not be aware that it has grown so much into your property.

welcome to GoY too

You have every right to cut off what is coming into your garden and, I would say, with a Russian Vine imperative you do so as they grow so quickly. Having said that do not fall out with your neighbour and seek their agreement. Always keep on good terms with your neighbours.

Russian Vine has a folk name - Mile a minute - because it grows so fast. Take action now.

The business of returning it comes from a case a long time ago where a neighbour harvested apples from an overhanging tree and sold them at a lower price than the tree's owner. The court ruled the overhanging branches still belonged to the tree's owner.

Unfortunately, if you don't tell your neighbour what you're doing it can lead to friction because they think you've just dumped the cut material over the fence.