I am writing an essay about children's right and somewhere, it is stated that all kids should have the right to primary school, children have the right to good quality health care, and so on. I looked up the definition of a right and I found that it's moral or legal entitlement to have or do something. I am living in developing country and we don't have a lot of infrastructure such as public school or public hospital. But according to the definition of a right, can a child go to a private school or private hospital without paying any fee and tuition(their parent cannot afford) just because that's their right and they are entitled to get the service?
The Universal Declaration of Human Rights
somewhere, it is stated that all kids should have the right to primary school, children have the right to good quality health care, and so on.
I suspect that the document you are thinking about is the "Universal Declaration of Human Rights" promulgated by the United Nations, and in particular Articles 25 and 26 which I quote below:
Article 25. (1) Everyone has the right to a standard of living adequate for the health and well-being of himself and of his family, including food, clothing, housing and medical care and necessary social services, and the right to security in the event of unemployment, sickness, disability, widowhood, old age or other lack of livelihood in circumstances beyond his control.
(2) Motherhood and childhood are entitled to special care and assistance. All children, whether born in or out of wedlock, shall enjoy the same social protection.
Article 26. (1) Everyone has the right to education. Education shall be free, at least in the elementary and fundamental stages. Elementary education shall be compulsory. Technical and professional education shall be made generally available and higher education shall be equally accessible to all on the basis of merit.
(2) Education shall be directed to the full development of the human personality and to the strengthening of respect for human rights and fundamental freedoms. It shall promote understanding, tolerance and friendship among all nations, racial or religious groups, and shall further the activities of the United Nations for the maintenance of peace.
(3) Parents have a prior right to choose the kind of education that shall be given to their children.
Some countries also put language like this in their constitution.
What Does This Mean In Practice?
So, what does this mean in practice, is what you seem to be asking when you say:
according to the definition of a right, can a child go to a private school or private hospital without paying any fee and tuition(their parent cannot afford) just because that's their right and they are entitled to get the service?
The short answer is, probably nothing, and certainly not a right to go to a private school or private hospital without paying any fee or tuition that their parent cannot afford.
These Rights Are Usually Not Self-Executing
Some rights in a constitution or statute can be directly enforced in court without any further legislation. Other rights are really just statements directed at lawmakers which lawmakers are supposed to try to carry out as well as they can if they take their jobs seriously and are sincere that can't be legally enforced.
The Universal Declaration of Human Rights, even if your country has adopted it (and most countries have) is usually not "self-executing", which means that the Universal Declaration of Human Rights can only be enforced if the government of a country passes other laws to implement it. It cannot usually be enforced directly in a court.
The Affirmative Economic Rights Are Usually Aspirational
Some rights are pretty easily to protect simply by passing laws telling government officials how to do their jobs.
For example, consider Article 11 of the Universal Declaration of Human Rights:
Article 11. (1) Everyone charged with a penal offence has the right to be presumed innocent until proved guilty according to law in a public trial at which he has had all the guarantees necessary for his defence.
(2) No one shall be held guilty of any penal offence on account of any act or omission which did not constitute a penal offence, under national or international law, at the time when it was committed. Nor shall a heavier penalty be imposed than the one that was applicable at the time the penal offence was committed.
If the government passes a law telling law enforcement officers and prosecutors and judges how to behave when people are charged with penal offenses, and if the government refrains from charging people with penal offenses when it is unable to comply with Article 11, it can implement Article 11.
But, implementing Articles 25 and 26 is not nearly so straight forward. As a result, most countries consider at least part of the Universal Declaration of Human Rights that seems to create an affirmative duty to provide certain government services, such as Articles 25 and 26, to be what is called "aspirational". This means that the country thinks of these rights as goals to try to achieve rather than morally or legally binding obligations of the country.
Governments Have Wide Discretion Regarding How They Provide Affirmative Economic Rights Even If They Obligate Themselves In A Binding Way To Make These Rights Available To Their Citizens
Even if a country passed a law enforceable in court making Article 25 and 26 the binding law of the country, and not merely aspirational, the government of the country still has wide discretion to determine how to go about taking action that will secure these rights for its people.
The Example Of Education
For example, most countries provide "free" education "in the elementary and fundamental stages" to most of their people by establishing taxpayer supported public schools, rather than by paying for their tuition and fees at a private school. Either approach would satisfy the obligation to provide a free elementary education, but it would be up to the government of the country to decide which approach to use and that wouldn't have to be the same for everyone in the country.
In the United States, almost everyone has a right to attend a public elementary school for free, but if someone has a disability that the local public school is not equipped to handle, federal laws in the United States give that person's parents the right to demand that their tuition at a private school be paid for by the school that would otherwise have had an obligation to provide that child with a free elementary education.
In Quebec, Canada, most people attend public school free of charge, but Roman Catholic children have a right to attend a school run by the Roman Catholic Church at government expense because there is language in the Canadian Constitution that says so.
The Example Of Medical Care
Similarly, there are all sorts of ways that a government can go about insuring that a particular person has a standard of living adequate for the health and well being of himself and his family, including . . . medical care . . . and the right to security in the event of . . . sickness [and] disability[.]"
One way to accomplish this would be to establish public health clinics and hospitals for those who cannot afford private health care. Another way to accomplish this is to require health care providers to provide services for free to anyone who can't afford them. Another way to accomplish this would be to establish a government benefit plan for people with incomes too low to afford care that pays for private health care providers to provide care. Another way to accomplish this would be to subsidize the cost of buying health insurance for people who can't afford the full price of health insurance, and then to have the health insurance company pay private health care providers. Another way to accomplish this would be to require employers to provide health insurance for their employees and parents who can afford to do so to provide health insurance for their children. Another way to accomplish this would be to establish a health care program that provides health care to everyone in some category of people, without regard to ability to pay, with tax dollars, by paying private health care providers whom people covered by the health care program are patients of. In the United States, a mix of all of the methods above are used to provide health care to its people.
Most countries (e.g. Germany) provide health care primarily through nearly universal provision of private health insurance, sometimes through a mandate to people who can afford it, and sometimes with subsidies.
Other countries (e.g. France and Japan) have single payer systems where taxes are used to pay for a single national health care plan that pays for private health care providers to provide health care services to its people.
Other countries (e.g. the U.K.) have systems where taxes pay health care workers to provide health care services to its people and most of those health care workers are government employees.
The bottom line is that even when the government of a country decides to turn a non-self-executing aspirational affirmative economic right to something in the Universal Declaration of Human Rights into laws that actually deliver these economic benefits to its people, a government has wide discretion regarding how to go about doing it.
Governments Can't Do The Impossible, They Can Only Do Their Best
Since no legal document can compel a government to do something that is impossible, something like a right to free education or a right to medical care, isn't very specific about the quality of the education or the medical care that the government is obligated to provide.
In a developing country where the government can't afford to provide very high quality education or medical care, most people would say that the government has met its moral or legal obligation to provide education and medical care if it has spent as much as it can afford to spend on these services given its limited national wealth and its limited ability to secure funds from a portion of that limited national wealth for these purposes through taxes or ownership of natural resources or whatever.
The Example Of Elementary Education
For example, in a poor country, free elementary education might be provided in very meager buildings or even out in the open, in very large classes with teachers who are volunteers or who are paid but don't have much in the way of formal qualifications to be a teacher.
The Example Of Medical Care
Similarly, in a poor country, medical care might include vaccinations, antibiotics and prenatal care, provided by minimally trained paraprofessionals like China's one time "bare foot doctors", but the country might simply be unable to afford expensive treatments for cancer or organ transplants or high technology MRI scanning, even if the government agrees that in an ideal world where it had enough money to obtain these things it ought to make them available to everyone in the country.
Whatever a government is legally obligated to do, it can only do in practice what is possible, and figuring out what is possible is an art and not a science, called "politics".