My daughter (7) is a farm girl, so she's been handling chickens and has even been responsible for feeding and watering the brooder chicks for at least the last two years. This year she did the primary socializing for the six new ones we're getting ready to turn loose into the main flock.
The biggest problem we've run into is allergies when they're fledging, all that fluff coming off is an allergy nightmare, but chicken diseases are largely chicken diseases and human diseases are human diseases. Once the chicks are outdoors, the allergies clear and next year I hope to have my outdoor work space set up with electricity/water so I'll be able to brood outdoors rather than in the laundry room indoors, which will be better for all involved.
Short of not eating around them and practicing basic sanitation, we always wash hands after handling any of the animals/birds, there's not much that wants to munch your chickens (feather lice f'ex don't want humans. We don't have feathers to eat, so we are an inhospitable host) don't want to munch on humans. If your birds get parasites - and it happens, they can come in with wild birds coming to eat the chickens food and leaving gifties, they are bird/feather specific.
We don't kiss on our chickens, or really even encourage picking them up unless we're socializing them as chicks or checking feathers/vents/feet/etc. We handle them/ hand feed often, but they really seem to prefer being left to do chicken things. Kids tend to drop them if they flap and I feel like the birds are more likely to be hurt by a kid freak out than the kids being hurt by the birds.
We don't currently keep roosters, but in the past, the roosters have made it clear that they are not here to be hugged and my daughter has respected that. Mostly, she got a good pinch on the backside once and came to the conclusion on her own that the hens are nicer.
She dropped a duck flat on it's butt once, and after that I encouraged her to let everyone keep their feet on the ground after that. Our ducks fly like bricks. Straight down.