Before the net and google and other search engines, these kinds of things would fall off the "public radar" over time and folks could get on with their lives. In that regard I can certainly see how and why some would be very interested in having that past light turned off, particularly if they were innocent ... and most reporting services rarely reveal that info, they've moved on to the next "horror" story and never followup with the charges being dropped or the person found innocent.
Then again, I don't like the concept of censorship as the original reporting was done in good faith (usually) and in the public interest.
Something other than censorship would be ideal, ie. for each related report a person does not like there is a report that shows justice was served and BOTH are returned for any single search... but that is a nightmare of coding and database use so I do not know what the answer might be.
I do know, however, that if this genie is let out of the bottle the web will become a very much different place, one where history can be rewritten by whoever ... and whenever.