well, you can sink 3m fence posts in the ground - go for 100mmx100mmx3m - you will need to have 60cm in the groun leaving you with 2.4m above ground - you can anchor this with a ready mixed post hole concrete or you can mix your own sharp sand - 4 parts - and cement 1part with any ballast or hardcore (bits of brick, stones etc. You can put the mix in dry and pour water in - about 8litres. You could fix chainlink or wires but bear in mind that you will need to be able to get to the hedge to keep it trimmed hard back so wires are probably better (and easier) than chainlink. I really like the wichurana climbers such as New Dawn, the barbier ramblers or there is a lovely bourbon, Kathleen Harrop which is a sport of Zephirine Drouhin and has all the advantages of that rose but is a softer pink. Both KH, ZD, New Dawn and Nadame Alfred can cope with a degree of shade and all have lovely fragrances.

It sounds like a maintenance mess to put a climber in front of an unruly hedge like that.

Might be easier to maintain if you first trim the hedge back hard and even on your side. Some types of hedges will regenerate from bare sticks and some do not, do you know what kind of bush it is? If it will regenerate you might be happy with it in a year or two after successive yearly haircuts.

If not, then why not plant some bushes on your side at the same height? You can mix in roses and evergreen bushes perhaps to hide it from view.

you are right about the fencing. If you go for 2.4m high closeboard and put a capping on the top and posts, It is a very lovely fence to have. It can be painted or stained and is sturdy enough to attach wires or trellis on the posts. It will also take considerable weight if you choose to grow a vigorous climber on it.
In the UK, we have had a lot of neighbourhood issues with the gigantic and fast growing leylandii conifer hybrid. Murder has been committed! We have now passed a high hedge law which entitles people to sue for removal of boundary hedges over 3m high. My partner worked as a fencer for a number of years - many, many vicious and acrimonious arguments centre on boundary divisions - sometimes over mere millimetres of space!

It is about the same over here, quite literally every feet of the ground is taken. Our neighbors generally are not bad, just they quite literally do not prune anything unless it tries to eat them. :))) Luckily it's the only thing on our border that is somewhat invasive (apart the grape on the other side, but it is quite good one, so I don't mind).

Thanks for the advise about the fence, I think we will try something like that. Bareroot season is almost over so I will stick with annual climbers (like sweet peas) for this season, so plenty of time for me still to pick a cute climber rose for it (or clematis, but since I already have 20 clematises in the garden... rose would be more nice).

Climbing Cecile Brunner gets enormous in my climate and will devour any fence. It's fairly shade tolerant. New Dawn seems wimpy by comparison and isn't really aggressive or flowering in the shade. Now if you really hate your neighbor's shrub, Cherokee will cover anything.

From your photographs of the hedge, it seems to be Prunus laurocerasus. This plant can be cut back very hard in April.
However, even if you cut it back extremely hard, it will still need trimming every year. If you plant roses and other plants too close in front of it, access will be difficult.
You could plant shrub roses at intervals in front of it, with tall herbaceous plants in between. These can be cut down in winter/spring, giving you some access to the hedge.
Whatever you decide to do, all the plants in front of the hedge will need heavy feeding as Prunus laurocerasus is very greedy.
Daisy

About feeding it is not very good news, but I suspected it might be something like that. Hm, maybe then I should dig in some plastic/concrete sheets there also, that should limit how much the roots spread to our side. We have some weird big concrete tiles laying about from the previous owner (1x1m), those could come in handy, since that would be a nice way to dispose of those tiles also.

Hybrid Musks? They're good spread out in a fan-like formation and have lots of branches and foliage. Nice scent, too. Most grow much bigger here in the USA than in their native England.

JMO, but that's a pretty hedge, a nice green backdrop for more roses. I would try climbers on rebar teepees in front of the hedge--you would notice the climbers, not the hedge.

I too would go with hybrid musks. They are shade tolerant, spread nicely and are not too thorny so it wouldn't be a hassel to prune behind them. The species rugosas are also somewhat shade tolerant and sucker wildly and might even compete well enough to hinder the other's growth. Or find some chocking vine to naturally kill off the hedge.

Roses on teepees might be a good idea actually and I do like hybrid musks also.

As for the hedge, well it looks better in the pictures than it does in person, but for me it really is a concern on a long run, since Prunus Laurocerasus is just not meant for reasonable hedges, it is something that easily can grow 20'+ to all directions and tends to bald near the roots, so eventually it ends up with tree trunks and 5 leaves, or a monster plant. I gave a thought to a chocking vine, but I can't remember anything growing here that could do that sort of thing (and do not take over whole garden in a process). On a bright note, it is at least good wind shield from the east-south winds, so it adds up to the warmer microclimate.