It crazy how many websites out there around bible collect information and share it with third parties. I looked at a bunch and couldn’t find anything that’s completely private.
Worst case, I would OK with them collect information but not share any information with third parties.
For me, something based around Sword is important, as it allows you to use a bunch of different Bible versions, commentaries, dictionaries and other works that you can freely download from various repositories. My app of choice is AndBible, and I recommend it wholeheartedly. I don’t know whether or not they collect any information, but I believe they do have a warning before you search for Sword modules, since accessing those databases can identify you as a Christian.
EDIT: No, sorry, the warning is on BibleTime, my desktop app of choice.