Hey

Yes student from private schools can take the exam..you can easily apply for various national level examinations.

Go on the official websites of different exams you want to take and simply apply.. Hope it helps thank you good f luck ...

Please explain your question properly. What do you mean by private student? If you are talking about private School students, then yes they can also give NEET or AIIMS exams.Or else if your talking about students who gives their exam only  i.e. who are doing non schooling, then yes they can also give these exams..