Government owned Aer Rianta used to run Dublin/ Shannon and Cork. However, following recent change in policy its was decided to split Aer Rianta into three seperate bodies to control each airport as a competing industry so that the smaller airports (cork and shannon) would receive adequate investment instead of it all going to Dublin...However they still remain Government owned. As for the other airports such as Kerry, Galways, Knock, Sligo and Donegal are all privately run!!!
Hope that helps you!