What privates are you interested in? What’s your budget? Where you go will be unlikely to matter for law or medical school.
Your college GPA, LSAT/MCAT and then medical activities for med school or possible work experience for law school will be what matters - not UC Berkeley or U of the Pacific.
So need to know your budget and what interests you in a private - size, location (urban, rural, suburban), weather, religion, etc.
What do you think you want to study? It’s not about getting into a school with a less competitive major. If you know (and you may not) what you want to study, then that’s what you should go for.
Your major, in my opinion, is more important than your school. You want to study what you love - especially to maximize odds for grad school - if you love it, you’re likely to perform better.