Hello @Ashley_Hobbs ,
Welcome to the Asana Forum 
I will assume that you are talking about Goals, as the title of this thread says “subtasks”, but in your text, you clearly speak about goals.
To this point:
They actually do by default
Everyone in the company can see all Goals and their sub-goals even if you have not added the person manually. By default Goals and sub-goals are public to the company. Only if you manually make a Goal or sub-goal private, then you restrict its view to members only.
Adding a person to a specific public sub-goal rather serves as a mean of creating visibility as to whom is supporting the achievement of that sub-goal and this is usually a manual decision to take.
Worth noting the “My Goals” section will only show Goals that are assigned to a person; if they need to see any sub-goals related to the Goal that is assigned to them, they will have to click on it first.
You can check out this topic if that’s what you are looking for, there we also share some tips, and you can upvote that request; the more upvotes the higher the possibility that the feature request gets prioritised.
I hope this helps. If not, please feel free to continue the conversation here.
Cheers,
Rosario