I want to create a user profile field that displays under the user's profile picture on posts, which is fairly straight forward with a template modification. However, there only seems to be two permission options: whether or not the member can set the value.

I want to take a more nuanced approach - I want this field, which is just a free text field, to be displayed only if the user is a member of certain groups. Otherwise, it should only be configurable by an administrator, just as all profile fields can be modified directly in the ACP.

Is there a way to do this out of the box or will I need to look at plugin development for this?

You would need a plugin to accomplish this level of granularity. 

Use this:

 

+1 because I'm looking to accomplish the same thing.

@Adriano Faria, can your plugin the previous post linked to achieve this? From looking closely at its description and screenshots, it seems like no. But if your plugin already does this or you're willing to implement it in an update, I'll be very happy to buy it!

You can choose who can use and who can view a profile field. That’s what it does.