The privet & Leylandi might have robbed the soil of any worthwhile nutrients a couple of feet either side of it.
Like has been said beech roots are a surface nutrient thief of anything useful for the whole diameter of the tree and a bit beyond either side of a beech hedge ?
Perhaps even consider to use a mini digger and drop in a three foot deepby one foot wide trench along the hedge line to break the roots then refill the trench , renovate & manure the beds well to kill off the feeders 7 make the beds workable.
Are you able to consider having a 4 foot wide slab or grass path by the hedges and moving the beds over .
This will not only help you wrt soil quality & light it will also allow you to trim the hedges and have an all way round access to the beds .
I have 3 foot wide paths made of 3 inches thick concrete all round my fence boundary.. It stopped the neighbours weeds coming in and also helped stop roots invading for all the paths are laid over a 3 feet deep by 3 foot wide bed of compacted concrete rubble or two inch crushed stone.
You didn't say if you own all the hedges , so I though you & a few others might find this useful .
In England & Wales
Don't worry about the neighbours . The law is on your side so long as you do things correctly .
As your neighbours didn't worry when they decided to say stuff you and let their trees & hedges cause you a problem . Legally you're entitled to cut any branches or sever roots on your side of the boundary so long as you do not cause a dangerous situation to develop /happen ( i.e. make the tree unsafe and likely to fall over onto things or people ) if they are unwilling to do anything about it . Cutting things back to the fence line ( do make sure you define it correctly ) and return the cut bits back onto their property in case they try to get you done for theft of things or offer them and see if they will allow you to dispose of them yourself .
My neighbour on one side grudgingly allowed me to cut & dispose of 12 foot of Leyandi branches growing on my side for 40 feet of my boundary line . I chipped it and used the chips as a drainage medium in a three foot deep trench .
The " neighbours " on the other side said I could not cut their rambling rose that was all over my adjoined garage and free standing shed.
So I waited till they went on holiday, cut it back then placed the bags of chippings on their door step with a copy of the county & DEFRA rules about hedges in a clear poly bag securely taped to the bag touching their door..
They took the hint, have never spoken to me again and got someone to cut their climbing rose down & dig it out.
If it's not your trees but a neighbours trees etc & it comes too close to your buildings you can also ask them to remove them at their cost as they are likely to affect your foundations .
I think there is also a legal requirement that can be enforced as to the height of a Leylandi or other type of boundary hedge ( possibly no more than 3 mtrs high on the owners side )
Your local authority online is the place to look for the up to date details. Possibly under the titles of nusiances and neighbour problems