Not directly applicable to RIAA here as they started this by issuing the takedown requests, but I often get around the Streisand effect by ignoring things and then issuing takedown notices later.

Like for example, if something came out that some persons might use to try an embarrass me, I let it stay out for a while and don't really address it. Two to three people tweet it and it doesn't go further, hundreds of people weren't compelled to save it "for the lulz". Then a few weeks or months later I get the service to take it down, the person that posted barely notices or understands that their voice is not strong enough and the bar to change that is much higher and they've moved on to some other issue de jour.

Yes, there is a possibility that it circulates pretty far initially, or that the thing being shared gets ingrained into the consciousness as truth, but in those circumstances providing a reaction involving the law or lawyers would still amplify it.