How can I make my lists private for logged in users only?

You have complete control over which lists to show where. Display of lists is done with short code and there is a powerful short code generator for your convenience too. So, you can create a bunch of Link Lists for public and assign them to one or more categories that you create and show in your public pages by calling only those categories. Assign all the Private Link Lists to another category that you don’t want everyone to see. Then create a logged in area or simply password protect a page and call that private category with shortcode in that private page.