Hi @Jamie8 
As it stands, it is not possible to make a project public but a task or subtask private. When somebody has access to a project, then they have access to all tasks, subtasks and conversations within that project.
There is an existing thread in the #productfeedback category regarding this topic. If you haven’t yet, I would recommend to add your comment and vote to it: Private and Hidden Tasks/Sections in Public Projects
Have a great Friday Jamie! 