Answer: Based on your description I am not certain why they would suddenly look thin and sparse. Perhaps a late frost damaged some of the foliage or there may be a seasonal disease or insect problem in your local area. You might want to check with your local county extension to see if they are aware of anything currently.
In general, pruning of an informal privet hedge or shrub would consist of removing a few of the oldest stems by cutting them off way down near the ground. This thins the shrub and opens the center to light and air. You would do this in late winter or very early spring, before it starts to grow for the year. But, you could still do it now. Take maybe a quarter or a fifth of the oldest stems every winter/spring. This keeps the plant in constant renewal.
For a formal hedge, you would trim it back quite hard every few years to stimulate branching and keep it dense. Do that hard pruning in late winter. Your regular maintenance trimming during the growing season each year would shear or shape it at an angle so the sides slope. The hedge should be narrower at the top than at the bottom. This allows sun to reach the lower part of the hedge and keeps it dense at the bottom.
I hope this helps.
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