It is a definite possibility that the duck made the chicken sick. Possibly Mycoplasma gallisepticum (https://en.wikipedia.org/wiki/Mycoplasma_gallisepticum). It is also always a good idea to keep dry feed dry (mycotoxins etc). Fortunately most bird diseases are not transmitted to humans (except of course the dreaded "avian flu"'s, not to be taken lightly).
I free range my duck with my chickens - he is a good guard animal - and I give the duck a small kiddy pool to play in, and the chickens get a nipple waterer. The duck is not very interested in the nipple, and chases the chickens away from his pool, so the cross contamination is kept to a minimum. Ducks being ducks I have to refresh the kiddy pool every day. The chickens roost on bars, so in the barn, they don't come into much contact with the duck poop. It has worked out OK for the last couple of years we've had the duck, no sickness here.