Earlier quoted context omitted.
An app downloading on behalf of the user isn't a privacy issue. Are their servers acting as a proxy? Whether that's a privacy issue depends on how they configure/limit/audit their servers.
An app downloading data on behalf of the user is basically what Cambridge Analytica was doing. The problem is that data ownership is complicated. If I know your phone number, can I share it with other people? That was CA (me downloading data about all my friends). Here the issue is private messages — is it okay for me to share the messages you sent privately to me? A lot of people will get quite upset if you do that!
That's half of it. Is the app sending the data back to the app makers? If not then it's extremely different.