Privet is the common name for a shrub called Ligustrum ovalifolium

In the warmer parts of SC, it might also be Japanese Privet (Ligustrum japonicum), aka Waxleaf Privet. It is hardy to USDA zone 7--if you don't know your zone, ask your local garden center, or the cooperative extension. Both kinds of privet are fairly fast growing, take well to shearing as hedges and topiaries, can eventually grow tree-sized if not pruned, and have oddly scented white flowers, and small, black, inedible berries if not pruned. Japanese Privet has darker green, stiffer leaves, is hardier to heat, and makes a more formal statement. Common privet is hardier to cold, has medium green leaves--except for variegated forms--and has a tendency to drop its leaves in very cold winters.

Sorry never looked at the origin of the questioner. We have a Privet bush which is neither ovalifolium nor japonicum. It is definitely deciduous rather than evergreen.

Maybe L. vulgare, Owdboggy?

Could be. It seems to be very common in this part of the world as a 'wild' shrub.