I think (if there's not already) there should be a sticky at the top, warning people that they cannot delete a thread and therefore if they want to provide sensitive information, maybe they should do it as a guest. Otherwise, if a mod can go into a post and delete quoted info that is sensitive, that is a better solution. Deleting the whole thread would be my least favored option.
Going off of this, maybe have people post as a guest if it's going to be a topic they'll want deleted (they could then still say this is so and so if they wanted to have people know who it is posting). Have a mod remove ALL guest posts, or all guest thread starts, after a certain amount of time, maybe 24-48 hours. That should encourage the guesters to register if they don't want their posts deleted and encourage the regulars to post sensitive topics without having to worry about it living forever.