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?
It's a bit difficult to give an estimate with the information provided.
I'm not sure if your question is related to adding privacy to an existing server code or you need the whole server side done. The later is significantly more cost and has an unknown scope based on only the info given.
You might also want to look at platforms like Parse and see if you could get all you need done there vs a custom server side system.
Happy to chat more on a call if you want to provide more info or have more detailed questions.
I currently host the backend of iOS apps for a client. I can forward you to the app developer who is also an iOS Development Guru (Recently featured by Techcrunch). I'm the host/Sysadmin for the backend of the apps but he will be able to answer the type of questions you have.
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.