The Mormons have plans to convert the entire pile of images into a searchable database -- in the next six months. I understand it will be available on-line for you and me to use free of charge.
This is no small task. They have thousands of volunteers that have been doing this for years, with other indexing projects, but they have been ramping up for this project, training new volunteers and getting ready to jump on it. The program they use downloads images a page at a time from their storage server, and highlights various parts of the image while the volunteer types the data into input forms. Each page gets sent to two different indexers, who transcribe the data independently. The program checks the data, and any discrepancies are sent to a third arbitrator volunteer who decides which data is correct.
So yeah, they're serious about this stuff. And although they try to maximize the accuracy, by the very nature of the amateur volunteers some mistakes are bound to slip through. If this project is anything like the 1930 census they already have indexed, you can still get to the original scans to check the data yourself. After you have found a record by searching the indexed database, if you have an account on ancestry.com, a click or two will bring up the scanned image of the page that name was on.
As far as Mormons being a huge part of the traffic, I expect the indexing server needs a copy, but other than that, most of them will be downloading the images from their server. I don't expect there are many Mormons requesting their own copies if the data.
As far as privacy goes, I've seen worse. I have a third cousin who is into genealogy. Somehow he got my family's personal information and stuck it on his web site--all about our mutual third great grandfather. One feature is a tree of all of his descendants. It has names, birthdates, family connections (mother's maiden name, for example) and such. After repeated requests to remove me from his list, we're still there. In contrast, the Mormon family search won't show you any information on people who are still alive. There's a good chance your data is in their database if you have a close family member who is Mormon. Just nobody can get to it until after you are dead. They seem to be making a reasonable effort to maximize genealogical research capabilities while trying to avoid privacy problems.