it's possible your chickens could pick up something from wild birds. Possible, but not all that likely. The most likely things are probably mites, lice. or worms. Those are unpleasant but not dangerous if treated.
It's possible the next person you meet could give you head lice, polio, measles, or the flu, but only if they are infected. If they are not infected they cannot give it to you.
Chickens can get parasites (worms) from earthworms, crickets, grubs, slugs, and other creepy crawlies they catch and eat. They can, it's possible, but only if those critters are carriers. Mine eat those kinds of things all the time and they don't have worms. I butcher chickens throughout the year. When I do I cut open some or the intestines to check for worms. Mine do not have worms although they regularly eat things that could give them worms if they were infested.
I feed outside in the run. Wild birds often eat that food. Sometimes I find a wild bird in the coop. Mine are exposed to wild birds all the time. So far mine have not gotten sick or picked up anything from that exposure to wild birds.
Mine eat mice, frogs, small snakes, and all kinds of other things. Chickens are omnivores that operate on the principle of "do lunch or be lunch". If it doesn't eat them first they'll eat it. Sometimes when I trap certain critters I cut them open and put them in the run for the chickens to eat. Mice they eat whole after a serious game of keep-away but other larger critters they need help opening.
My chickens are healthy. They have never had mites, lice, worms, or other diseases. Coccidiosis is in the soil but I manage that. Mine are exposed to life and get to act like chickens. It's possible I could fine a problem today, life is like that, but so far it has not happened.
I suggest you do as Folly said, get rid of the carcass if you don't want them to have it and get on with your life. You should continue doing what we all should be doing whether you've seen them eat something or not, observe your chickens for symptoms and regularly check for mites and lice. It's a good question but I don't think you have any reason to be that concerned. The odds are very high your chickens will be OK.