iOS Developer and Co-Founder of One Month Apps. Previous employer was Quicken Loans where I was the Sr. iOS Engineer on the mobile team.
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.
There's usually a few downsides to developing with any kind of cross-platform solution.
1. You won't be writing apps in their native languages (ie. objective-c for iOS and java for Android). Instead, with Corona, you'll be writing your apps in Lua, which is a scripting language.
2. A lot of times not writing apps natively can negatively impact the performance of an app. I can't speak for the Corona SDK directly, but using 1st-party tools and languages properly should always yield the best app fluidity, responsiveness, and performance.
3. You will be directly limited by the Corona SDK. What this means is, when you are developing your apps, you won't have the luxury of being able to use the latest and greatest technologies. You will have to wait for Corona to add those specific features into their own library, before you'll be able to gain any benefit from them.