I've got DropBox working in one of my apps, but I haven't played with public access. I've got it working where user's access to the folder created by my app requires an invite from the owner of the shared Dropbox folder. Much easier for a single user as each device would only need a 1 time sign in to their existing DropBox account.
This would work for sharing data across devices or users with permission. The DropBox account isn't created by the app, the user must create an account by applying to DropBox directly first, so it isn't a seamless data share. It does the job for may app though.
In another one of my apps, a reservation system, I use a server based mySQl database where the user creates an account and password to gain access and the app will download the json file or data from the database. I struggled with the mySQL/PHP, I'm terrible, but have hacked out about 25 scripts to run the app. This is the way to go IMO, unless the server space goes away and so does the users data.
I use DropBox for the one app because the user's data is important, private and in their control, I also don't have to pay for the file storage.
Neither method is "quick and easy" though, just say'n.
H wrote: "Specifically, I want this for cross-device reading - not user file storage"
I have to admit, you lost me with this one though. I was assuming you wanted a user's data to be shared across their devices. That would be the definition of user file storage. IMO.