The term "privet" applies to a large variety of small shrubby trees that make good hedges, and mostly come from the plant genus Ligustrum. However, many are aggressive and spread quickly, and have fallen out of favor for hedges because they require such maintenance. Some locales even ban the sale or planting of privet, as it can take over large areas of wild land. There are other shrubs that look like privet, but are less invasive, easier to maintain, and in some cases may even be native to your area.
Gray dogwood and red osier dogwood may be good alternatives depending on your part of the country. Gray dogwood is a smooth-leafed shrub that gets very thickety, responds well to pruning, and likes moist and rocky soils. It resembles privet especially in its small white flowers, and prefers temperate climates. Red osier dogwood is a more bush-like shrub that has redder stems than privet would, but looks like privet as far as foliage. It is hardier than gray dogwood and thrives in more northern locales and richer soils. There are also several other types of dogwoods that make good substitutes for privet depending on your area.
Fragrant sumac, highbush blueberry, inkberry, deerberry, strawberry bush, and nannyberry are among the berry-bearing, non-invasive and pleasant planting alternatives to privet. They are closely related to privets and have similar small leaves, especially highbush blueberry. Some of these also have the side benefit of producing tasty berries to eat in the growing season. However, note that not all of the recommended plants above give edible berries, and remember that berries are likely to attract birds and other wildlife. This may or may not be desired.
There are a huge variety of native hedge-like bushes, some of which resemble privet more than others. New Jersey tea bush is one that looks a lot like privet, but is not as invasive. Others include false indigo, Virginia sweetspire, shrubby cinquefoil, maple-leaf viburnum and other viburnums, arrowwood, and black haw. For shaded locations, try boxwood, a close relative of privet. If you want a specific kind of hedge shrub or bush for unusual soils or climates, consult your local plant nursery or agricultural extension agent for recommendations.