I'm in Austria.
It's a contributions based system here which means you have to be paying in to receive health care. I am self-employed and have to pay 27% of all earnings towards health insurance and pension. I don't know what proportion of the 27% is health and what is pension.
There are several different Krankenkassen (the health insurance companies) and you are insured with different ones depending on your job - eg. I'm with SVS, but there are separate ones for railway employees, teachers, doctors, general ones for each Bundesland.
The SVS is a bit shit to be honest as I have to pay 20% excess when I go to the GP or for some outpatient treatment. There is no excess on any hospital stays and the treatment for that. There is a cap per year so if you had a condition which required a lot of GP visits and outpatient treatment you would pay the excess only up to the point of the cap and then no more. Some of Krankenkasse have an excess of 10% and some have none. You can't choose your Krankenkasse!
If you are unemployed your contribution to health insurance is covered as part of unemployment benefit, however you really do have to cooperate with them with regard to looking for work and accepting job offers. If you do not and they stop the unemployment benefit, the health insurance also stops.
Everyone has to be paying in, including pensioners who have to pay 5% of their pension towards health care. They do not have to pay excess however as they are insured in a different Krankenkasse to mine.
It's all very complicated, but the healthcare system is very good and it's easy to be seen and receive treatment. GPs are very thorough and you'll be sent for tests for very minor things - probably wouldn't happen in the UK.
One thing I do like is that you can see a Kassenarzt (has a contract with the Krankenkasse, you don't have to pay anything but in my case I receive a bill later of 20% of the cost due to the excess). You can also see a Wahlarzt (an elective doctor) in which case you pay upfront and then submit the bill to the Krankenkasse to get the money back (the amount they would pay a Kassenarzt - the Wahlarzt might charge more so you'd have to pay that). Or you can see a private doctor and again, submit the bill to the Krankenkasse and get some money back.
My dentist is a private dentist in Hungary. I can receive treatment there, submit the bill to the Krankenkasse and usually within 3 - 4 days I get some money back - not the entire amount because they will only pay the amount they would pay an Austrian Kassenzahnarzt - eg. the equivalent of an NHS dentist.
This sort of thing does relieve pressure on the dentists here as people can choose to go private and still receive a contribution from the Krankenkasse.
There is no problem whatsoever in getting a dentist appointment and treatment here.