Private matters may include family matters, sex, drugs, and anything that is difficult to talk about
My opinion is that college applications are not the place for that. You want your application to reflect how YOU will be able to be an asset to that college.
Go see a therapist or write on an anonymous blog or keep a diary. Leave it out of your college app. Your goal is to paint a positive picture of yourself that makes them want you on campus.
If it is very meaningful to you and your academic achievements and goals to talk about a hardship you have overcome, then speak your mind. I think people know when a writer is appealing to sympathy vs. evoking triumph over adversity.
If it is along the lines of a hardship you faced, because of illness, divorce, death, tragedy- and how it impacted you, Yes. If you’re going to wash your dirty linen in public, well, anything you wouldn’t say to a room full of strangers, you don’t say ANYWHERE, period!