Right-clicking on a folder and choosing the tab Permissions you can see the owner of that folder and the permissions applied for the owner, for the group whose the owner is member and for the rest of the world.
Assumed that, we can say a shared folder is a folder owned by a user (typically contained in his home folder) whose the owner choose to share the content by applying tolerant permissions (reading or even writing) to other users, better members of a group rather than all the world, for security reasons.
Instead we can say a folder is public if many users (some groups or all the world) have read and write access to that, no matter where this folder is or who is the owner, but it is assumed we are talking about one common folder. In this scenario the public folder owner is typically the administrator and the so called "sticky bit" permission is applied to the folder, that is everyone can create files but can delete or move only owned ones, to prevent chaos.
In a new Mint installation I see the Public foder has the same permissions of the others in your home... I don't exactly know its purpose...
To answer you question, I have to understand your needs: do you want simply re-create the folder as it was or do you need to have a unique folder shared for the three users ?