It's the other way around. Chickens, having been domesticated for so long, are immune to many diseases but can still be carriers, putting smaller, less domesticated birds at risk. Some chickens (especially larger/non bantam breeds) will attack and eat small birds (or rodents) in the confines of a run.
It's best to keep aviary type birds in their own space, caring for them first before moving onto your chickens to lessen the chance of transferring diseases to your less domesticated birds. Or keep quail instead of chickens as they can live quite safely with aviary birds.