Yes as long as they "give you back" the chopped off bits. Perhaps they are just lazy gardeners.

It's a shame sometimes when we think we are doing something that will make things easier. To placate someone or to cover an intrusion that we have no control over.
Snoopdog has the right idea, I would interplant the vine with over climbers, clematis, morning glory, honeysuckle,
whatever. If they smell or notice something really nice, possibly there will like it and ask ' thats nice, what is it '
Just a thought.

If you mean that you planted the Russian Vine between the two fences, then technically you planted it on their land as I would assume that the original fence would mark the boundary line between the properties. The "no mans land" as you put it would still belong to your neighbour regardless of them erecting a fence further into their boundary line.

If you planted the vine on your property (your side of the original fence), and the trellis is attached to it, then there's probably not a legal problem and as you said, they have the right to trim it if it encroaches on their land. However, as Snoopdog and Littleleggs put it, it might be a good idea to try and speak to your neighbours smooth things over and nip it in the bud. (Pardon the pun).

I have a 23-year old Russian vine - I find it flowers better on old wood but that it does get out of hand after a year or two. I chop mine back to the main stem every other year (3 years at most) just as new leaves are coming through - it's inter-planted with honeysuckles and a rambling rose. The RV leaves start a few weeks before the others so I know what I'm cutting back.

Thank you all for your comments. Yes the vine is planted on my side unfortunately it is difficult to reach over the boundary fence to trim as this is where I added the trellis. It is flowering now so don't want to chop it back yet but will do once it stops flowering.
Does any one have any ideas as what I can replace it with, ideally I would like something that is fairly fast growing and that would hide the 2meter fence. I have tried several clematis only one survives but it is very spindally, I already have honey suckel and ivy so would prrefer something else.
It needs to be quite hardy as I am on top of a hill by the sea in Sussex so I get the full force of the wind. It would have full sun from midday

If you have the time and energy I'd stick with the RV - it really can be stunning when it's in full flower. It just needs a firm hand to keep it in check.