My tendency is to try to make everything private if possible. This keeps object boundaries as clearly defined as possible and keeps the objects as decoupled as possible. I like this because when I have to rewrite an object that I botched the first (second, fifth?) time, it keeps the damage contained to a smaller number of objects.
If you couple the objects tightly enough, it may be more straightforward just to combine them into one object. If you relax the coupling constraints enough you're back to structured programming.
It may be that if you find that a bunch of your objects are just accessor functions, you should rethink your object divisions. If you're not doing any actions on that data it may belong as a part of another object.
Of course, if you're writing a something like a library you want as clear and sharp of an interface as possible so others can program against it.