This means that what happens in a person's home or private life is their business and should not be subject to outside interference.
In the above sentence, is the word "their" correctly used?
Yes, the use of "they" or "their" for a single antecedent whose gender is unknown has a LONG history and (despite what you will hear about people crying over the degrading standards of English) not a new invention of those who insist on gender equality.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
Welcome back after your long absence, bewise.
Please note that a better title would have been A person's private life is their business.
Extract from the Posting Guidelines:
'Thread titles should include all or part of the word/phrase being discussed.'