Traditionally, this would be expressed using "who". For example:
I'm the kind of person who doesn't really share much about my life.
That would clearly be correct.
But usages change and new idioms develop. If this usage is indeed becoming common, then it can be considerd newly correct.
However this Google Ngram chart shows very small use for "the kind of person where", although the use does show as increasing.
Google books does show such uses as:
- But I think Justine's the kind of person where, if she wants to read a book, she just goes ahead and buys it. [Where There's a Will by Elizabeth Spann Craig (2021)]
- If someone is looking for a patsy, that is the kind of person that they would pick: innocent, isolated, timorous; the kind of person where if you bully them they keep quiet about it; [Decoded: A Novel by Mai Jia (2014)]
- You become the kind of person where I say, we need more practical types like Myla Lewis in the world. [Clockwork Igni: Action. Adventure. Fantasy. Badassdom by Christina Bauer (2021)]
- So me, I'm just the kind of person where I'm always looking for the next. What's the next future? [Public Enemy: Inside the Terrordome by Tim Grierson (2015)]
- But Stephin has always been the kind of person, where, if you ask him what he thinks of the Spinanes record, he's going to tell you. [Our Noise: The Story of Merge Records, the Indie Label that ... by John Cook (2009)]
There are many pages of similar results. Is this enough to call it good current usage? That is a judgement call. I would prefer to stick to "who" in this construction. I don't see any gain in using "where", or anyadditional nuance or clarification. But I am not willing to say that it is simply wrong, as I once would have. However, I would certainly not use this "where" construction in formal writing, unless I were discussing the usage itself, or quoting those who use it.