Thanks for the suggestions. I forgot to mention the berries were spherical and always shiny, even when black, so that rules out both Capsicum & Viburnum species shown. I think I would have remembered if the berries were purple at one point, too, so that mostly rules out Capsicum. I did find a pic of (supposedly) Solanum nigrum online that had red berries, though there were conflicting captions on the photo. Maybe my mystery bush was Solanum nigrum after all, or maybe some other species of Solanum. Here is a link that might be useful:

I had considered a deciduous holly, too, suja, but the leaves and the shape of the shrub didn't match up. I'm afraid it is autumn olive. I had also considered this, but the leaves didn't look quite right in the pictures I had looked at. After reading sam_md's post, I looked at some more pictures, and the berries are a perfect match, right down to the little brownish spots on them. I checked while I was out walking today, and the undersides of the leaves are indeed silver. Mystery solved, I guess. Too bad. It's really a very handsome plant. I've always read it was a very invasive, undesirable plant, but I can see why it gained popularity when its planting was encouraged. Unfortunately, this particular shrub is located on private property, so I won't be able to chop it down. At least now I'll recognize it if it shows up on my property. I can add it to my list of things to eradicate, along with oriental bittersweet, mile-a-minute vine, garlic mustard, . . . The list never ends. Thanks to both of you for your help.

Yes, apologies for the tiny photo - it was either a huge file or a tiny one! I have followed up on your suggestion and my mystery plant does indeed seem to be Asparagus densiflorus. Thanks for your help.

Friends have small children! Must know what these berries are because small children make poor decisions! ==>>> fill small snack baggies with goldfish crackers ... tell the kids NEVER to eat anything in the yard... and to bring the baggie back when its empty.. for more ... its much easier to teach the crumb muncher ... as compared to denuding your entire yard .... most likely ... you will never remove everything in the yard that might hurt a child .... heck.. even if they eat enough grass ... they will be trotting around the yard ... but you are right.. these types of berries are what is called in the law .. an attractive nuisance .... they attract attention ... ken