Translation:It is not private, it is personal!
I have been following this discussion for ages because I found it interesting, but I forgot to mention I actually have an answer.
In Jane Austen's novels it is possible to make a distinction between a public sphere where everybody meets and a private sphere at home for family/close friends. This could explain the context of this sentence. Someone could say "you can tell me, we are in private right now", to which the other person replied "it is not private, it is personal" in the sense of "it is not something I share with family".
The content of Duolingo's sentences doesn't always make sense though so it's just as likely (or even more likely) that they mainly based this sentence on what is grammatically possible.
Private is something that is not available to the public, for example the terms of a contract or the meetings of a group or society. These can be confidential or in some cases even secret. Personal relates to one's "person". For example my photo might be available to the public on my website, but it is still my personal picture.
Yeah, that's fine, whatever. Not possible to tell from here, of course. There are ways around that, if it matters to you. You can copy and paste from elsewhere, or you can check this out: