I would say that it's mildew. Privet is prone to this condition and a dose of Bayer Garden Fungus Fighter Plus should solve the problem. You could try a mixture of 40% milk and 60% water if you want to go down the chemical-free treatment route.

Thanks for that Jimmy Theone; I did ask the woman next door, who is an expert on gardening (I do not see her very often) and she looked at the leaves at the foot of the privet, and said it was not dying, but reckoned it had been burnt by the sun this year, as the sun was so hot; this gave me hope. Even so, do you think I should treat it as you say, just to be on the safe side?