The European insurance card proves that the holder is entitled to the services that would be given to him for free if he was an insured citizen of the country where he is asking for medical care.
This can be very tricky, as everybody is expecting to get the service he is getting at home, but this will not always be the case.
For instance in Germany 'Kassenpatienten' and private patients can go to the same doctors and hospitals. As out-patients they get the same service, the doctor gets a different payment.
In Hungary the situation is different: there is no private insurance for Hungarians. The state insurance is valid only for doctors and institutions within the system, which offfers free care and gives services in according quality and availability.
When Hungarians decide to look for better quality, free choice of doctor, or consultations with appointments instead of waiting, they go to private doctors. In this case they know that the fee will be paid by themselves, and nobody is going to reimburse them.
If you are a European citizen you have the rights like Hungarians. You do not have the right to have non-urgent problems treated, but in acute situations you will be treated by state employed doctors and state owned hospitals.
In short: if you wish to be seen by private doctors you will have to pay, as nobody else will if you do not. Citizens of some EU countries will still be able to use the invoice and send it to their compulsory insurance. Germans, French, Austrians will have to send in their invoices, and they will be reimbursed according to national rules.
If you do not have private insurance you will be wise to find out in advance how your insurance is going to react to getting private doctors' bills.