The healthcare system in Poland - as it is in all of the EU - is based on equal treatment and equal access to medical services for all. The main state body responsible for the management of public funds for healthcare, and the pillar of the entire health insurance system, is the National Health Fund, or otherwise known as Narodowy Fundusz Zdrowia, NFZ.
As by the law foreign students have an option of either insuring themselves via NFZ or staying on the territory of Poland based on their international private insurance (usually received when applying for a National Visa type D).
Remember, in Poland having health insurance is obligatory. When applying for a visa to Poland and later on legalising your stay in Poland, you always need to present an insurance coverage document.
No matter which insurance you have, in order to be able to visit a doctor you must first be registered in a clinic - whether it supports NFZ or not.
In order to register in your clinic of choice you will have to provide your passport and your student ID card. They might also ask you to provide your PESEL number however if you do not have it issued you can present your international passport number instead.