Brutal pruning is required. Wait until Spring, mid March, then cut the entire hedge down to about 6 inches high. Feed and water to get good growth and within a couple of years it should look good again.
you could cut reduce the top down to the bare stems, privet regenerates well in my experience.
there's also a techinque known as notching, you find a node or bud on the bare stem, take a small notch out above the node/bud. This stimulates the bud to grow by disrupting the auxin flow in the stem.
We had the same problem when we moved in, cut it regularly, you can hack the height down as much as you like it will just grow with more vigour, then feed and water it to give it a boost. The bare legs will grow new growth.
One other tip. When you cut a privet hedge make it slightly narrower at athe top than the bottom, if you do this the bottow stays bushy.
I cut my hedge to have bare legs at the bottom for a while. If you want to keep the height, it isn't totally necessary to reduce the height, if you trim back any growth on the bare legs two to three times each year it will grow back although it does take a couple of years for it to return back to a lush hedge.
Cuttings also take well. These can be taken in the autumn but again take a couple of years to grow to a size when they can be planted out to fill gaps.
The hedge will also respond well to a good mulch of well rotted muck either in the autumn or spring to encourage growth.