Are we talking about the hedge being a leylandii ?

No it's a 6ft privet hedge that runs the whole side of the garden.

Ah gross feeder, doubt you will get anything to grow near it!

Depending on the aspect, you could plant out quite a few pernnials, but obviously not right next to the hedge, create a border and plant useing the following, Aquilegia, foxgloves, crocosmis, lysmackia punctata, lysmackia clethroids, achilleas, sedum spetabile, euphorbias, euphorbia characias wulfenii does rather well in this situ, asters, may need a little watering if too dry, alchamilla mollis, cerastium, iberris, lycniss, you could opt for certain shrubs for this situation, try and create a border and plant aprox 3ft from hedge, giveing yourself room for clipping etc.

I had a house with a privet hedge with Dicentra formosa growing under it once.

You could have a look under "D" at the bottom of the page and look at the "Dry Soil" pages of Goypedia. A lot depends on whether your dry side of the garden is in sun or shade, or a mixture.

I find Cyclamen hederifolium and C.coum both do reasonably well in dry soils provided it's shady.

Make a path alongside the hedge and plant a border on the other side of the path so its not so close to the hedge and that way you will have access to both sides of the border and not worry so much about dryness!
You will also be able to cut the hedge without treading all over your border plants.
Just a suggestion....:))

Good suggestion Pam

Thank you everyone I will pop tp the garden centre now with a list!! (",)

Look at Long Acre - 'Plants For Shade' this site specialises in the plants you're needing.