Hello Nanako,
I am answering based off how the words feel to me as a native US English speaker - not on any textbook definitions.
To me, a "private person" is someone who mostly "keeps to themselves". This tends to imply they are quiet and value spending their time alone.
While it has a similar meaning, a "loner" sounds more negative. As in someone who does not enjoy being around other people or has a difficult time being with other people. It can also imply that others do not like to be with them, thus forcing them into the status of being a "loner".
For example, a private person might enjoy their alone time, but also gets along well with others in situations where they need to be social or choose to be. On the other hand, a loner implies they do not enjoy spending time with others at all or that they, for some reason, cannot. Such as social awkwardness.
Either way, it is more polite to describe a person as a "private person" whereas a "loner" sounds more negative.
Hope this makes sense.