Cut the main trunk of each mature privet shrub down to 4 or 5 feet in height and remove all the lower branches. For smaller hedges, cut the shrubs back to where only a single, sturdy trunk remains, no matter the height. On young privet shrubs, simply cut the plant to the ground. The important point is to leave a trunk that can be used for leverage to remove the taproots of privet in older, established hedges.
Dig into the soil in a circle around the trunk of the privet, slicing through the lateral roots with the blade of the shovel as they are exposed. Where larger roots are encountered, use the blade of a mattock to cut them. Continue working around each privet, cutting the roots and digging deeper until all the lateral roots have been exposed and cut.
Push on the remaining trunk of the tree to see if it gives. If it does not give at all, continue digging deeper toward the main taproot that grows straight down from the center of the trunk. It may be necessary to remove more soil from the side of the hole to make enough space to dig at an angle toward the taproot.
Continue the combination of digging around the taproot and pushing back and forth on the trunk in all directions. Eventually, the tap root will break and the stump and root mass can be removed. If needed, use the mattock to make the final cut to sever the taproot from the ground.
Haul away all the debris and return the excavated soil to the holes where each shrub was removed.