Is there a real need for there to be two fences? You're always going to have problems with weeds growing up in that No Man's Land between them as birds will sit on there and drop seeds into the gap.
What are the neighbours like? Can you approach them and warn them that you're going to take action against the brambles in both your interests?
If you removed your fence is their fence in good enough condition to be the boundary - or is yours the proper fence and theirs has just been erected as they wanted more privacy?
What else have you got growing around there? If you need to spray I would suggest that Brushwood Killer is the stuff to use, but it will kill everything that it touches!