The answers depend on your state of mind when coming to stay in Greece. See the BritsinCrete page on Crete Wannabes which will give you an insight. Greece is Greece not the UK. Some things are the same and some very different.
I have always found the public hospital system very acceptable whether in an emergency or as a referral patient. Generally if people have a problem they seek hospital outpatients first. Unlike the UK there isn’t in general a GP system and certainly not where my home is. The village is visited by a government doctor once a week. There are private clinics which handle emergencies (ie very much the case in point in the tourist season). Private healthcare is staffed by many doctors who play the dual role of being public doctors and having separately their own specialist private practice.
If you speak Greek, of course one less hiccup, if there is to be one.
To sum up, there are many differing experiences, but overall, depending on where you live in Greece they do their best. As with anywhere there are heroic stories. Greeks are caring. Maybe other members have had other stories to relate?
Here is how to get more info on questions raised and related experiences of BiC members (if any) …
In the ladies board, you can find specialist medical topics. (You can only access if you have checked the "Ladies" gender box in your profile).
For other results, try using the forum’s general search covering one year or longer. This you do by clicking on the search button above any post with key words such as "health" "healthcare". The results will be comments in threads that may answer some of your questions. Do the same for things like "police" and "fire".
I have spent time in Sitiea hospital and could not fault a, the treatment and more importantly the speed of treatment being an ex. NHS nurse the service I had was second to none.
The only problem I have had is the amount of paper work and because I am covered by IKA you can only see these docs. to get the right stamps so your care is free. Not speaking very good and little Greek has never been a problem.