Good question. Do you know what kind of privet, and were they intentionally planted? Can you post photos? Also, what is your general location?

The only sterile (non-berry producing) form of ligustrum is L. sinensis 'Sunshine'. The only other reason a ligustrum would not produce berries is lack of pollination. As these are primarily insect pollinated, weather could very well have an impact on that.

Well, lack of pollination doesn't quite fit because, as I said, there is one tree in the stand that is producing lots of berries, and the others are not producing any. Pollinators would have to be unusually selective. These are in sort of a hedge at the periphery of someone's yard. That being the case, I might assume they were planted long ago. I have to suppose they carefully planted a sterile variety, and someone at the nursery screwed up and added a non-sterile one to the pile.

'Sunshine' ligustrum has bright yellow leaves and doesn't flower at all. To my knowledge, it is the ONLY sterile form of ligustrum.

In addition to Sunshine, there are a few other privets advertised to be sterile, or nearly so, but I have no experience with them. Sunshine is a sport of L. sinense, that was patented in 2007 and (as gardengal noted) doesn't produce flowers and is therefore sterile. Suwannee River is an older hybrid (L. japonicum x L. lucidum) grown in the South that is supposed to be fruitless. Golden Ticket is a complex hybrid patented in 2014 that has "low fertility with limited fruit set". Kindly is a recent introduction (patented 2024) that is supposed to be seedless (or nearly so). I don't think any of these varieties are entirely cold hardy in the north, so if the OP is in a colder zone it's probably doubtful that we're dealing with any of the sterile cultivars.

Well, there certainly isn't any Sunshine in this stand. No yellow leaves. I do understand that shade reduces the number of flowers, but doesn't eliminate them. But as I said, these trees all get plenty of sun. I am in Texas, so cold-hardiness is sort of irrelevant.

Photos might help. Does the fruiting plant's foliage look the same as those that aren't fruiting?

The patch is a mile or two away, so I can't run out and get a photo. But all the foliage looks the same for all the trees. What was striking was that in this one stand, there was just a foot or two of branches with berries.