FYI if this helps anyone:
I shot a quick email over to the developer asking for more information about some of the permissions, specificaly the "your accounts" and "your social information" permissions. The developer was very quick to answer my questions and for information here are his responses:
The "your accounts" permissions are for signing you up for Noom updates, but we'll only actually email you if you consent in the sign up flow.
The "your social information" permission is for inviting your friends to Noom Walk. Again, we won't actually act on this without your permission.
The "your system tools" permission is just for saving temporary app files.