No, it doesn't really depend whether it's public or private campgrounds.
It has more to do with where the campground is located and the crime that occurs outside the campground. If you are in a campground close to a large, metropolitan area that has high crime, the boundaries of the campground are not going to keep the crime from occurring there. If you are camping in a region that has a low crime rate, chances are that you will be okay leaving some of your accouterments visible when you are out and about.
That still makes it a toss-up about whether you need to lock your stuff up.
We pretty much leave out our chairs, table, etc. We don't leave out most stuff that can easily be "walked away" with, such as lanterns. We keep all of our electronics safely inside the RV - they are usually put in one of our lock boxes.
That's just my view.