Privet (Ligustrum) are susceptible to several leaf spot fungal issues and also root rots. It looks as though the plants are crowded which reduces the air movement and allows the spores to grow and flourish. I am not able to tell if the symptoms are caused from a root rot or a leaf pathogen.
I have attached links to additional resources that will help you manage the problem. A fungicide application (you may need more then 1) will help to protect the leaves that are not yet infected (Bayer Advanced, Ortho, Ferti-lome and Bonide all have broad spectrum fungicides that will work). The links will give you management tips as to pruning and removing the infected leaves that have fallen to limit potential re-infection.
Thanks for the question.