Answer: Several fungal diseases can attack privets, as can some destructive insects. The spots may be the fruiting bodies of a fungus, but I can't tell without looking at them. Why not collect a few of the affected leaves, put them in a plastic bag that you can seal, and then take them to a nursery or to your local cooperative extension office for a positive diagnosis? Once you know what you're dealing with, helpful folks at the nursery or extension office can help you decide which control strategies you can employ.
Best wishes with your privets!
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