The goal is to provide full privacy for users so it would be impossible to understand who is the author of content, his location, preferences or wether two users have communicated. How much would that cost and who is capable of doing this kind of work?
This can be handled fairly easily. The backend that accepts the content from users would not track any data related from the transmitting party.
From your post I can't tell if you'd still expect there to be user accounts or not, but to even provide an extra layer of anonymity, avoiding user accounts all together would be a good path to head down.
There are BaaS (Backend as a service) solutions that allow mobile developers store data in the cloud very easily. Some that I would recommend taking a look at are Parse (www.parse.com) and Azure (http://azure.microsoft.com/).
All that being said, without knowing any of the other details of what your app is supposed to do, it would be hard to provide a cost estimate. But I'd guess it would be on the lower-side of apps that require a remote backend.
If you want to talk more about this in greater detail, don't hesitate to set up a call.