I don't think it will do much harm - you should only really be concerned if they were chocolate coated doughnuts.
Excess sugar can trigger eczema in dogs, but the odd doughnut won't be a problem. If you see any unusual symptoms, the best advice would be to make a trip to the vets - but until your dog starts acting very abnormally, I wouldn't worry too much.
Just out of curiosity, was it a police dog? Everyone knows they love doughnuts!!!
nothing will happen all of a sudden if your dog eats glazed donuts. The
quantity at which your dog tastes glazed donuts do matters. If it is one or
two, then just serve your dog with one or two glasses of water. More than one
or two donuts may cause long term health issues like pancreatitis and diabetes.
If you find your dog has eaten too many choco donuts, then please do call your
vet immediately!! Never allow your dog to eat the table scraps with too many
donuts, even if it is glazed!! Avoid donuts as much as possible!!
Rather stop him from eating sweet things. Not good long term.