How to create a public and private space in confluence

The most simple approach is the one you suggest, but you don't explain what the problem you are having doing it is?

Set up space 1 with anonymous view access, and space 2 without it.  That gives you a public and a private space.

If you want to do this for parts of spaces, this is also possible.  You need to make the space anonymously accessible first, then you hide parts of it by adding "view restrictions".  The restrictions will cascade down to child pages, so you could start with a simple "page 1 (and children) public, and page 2 (and children) private", but you can put restrictions on at any level, not just the top level pages.