Where did all this come from?
"Privatizing" Social Security doesn't have anything to do with corporations, it has to do with making your account actually yours. Instead of it just sitting there behaving according to the whims of that body known as Congress, you could decide that you want your account to, for instance, ride with the mean return of US industrials, or stick it in a prime-minimum certificate of deposit, or invest it across all US pharmaceuticals, or sink it into Las Vegas casinos, or....
Okay, probably not that last item, because there would have been restrictions on what sort of risk you'd be allowed (though Vegas is hardly high-risk, except for the people who lose their money there). So the only "stockholders" would have been the people who are now -- us, with our accounts. The only "CEOs" would have been the people we call the Social Security Administration -- same for a bureaucracy. The only difference is that you would have been able to go online and look at all the charts showing your history and your balance and projections, and with that there would have been a button to click saying "manage my account". Click on that, and you would have gotten to a page showing investment options, and an amount you could invest yourself -- equal to not quite 5% of your total account.
And once you were receiving Social Security payments, you could keep doing that. And the beauty would have been that if when you died there was a balance remaining, your heirs could have gotten it (or you could have donated it against the U.S. debt or left it to bolster the health of the SS fund).
That's all that privatizing meant -- that you'd be able to control a teensy little part of your own account, if you felt you could get a better return than the government was (not hard -- even just calling Edward Jones and telling your agent to handle this money for you would have done that, for the entire existence of Edward Jones).
I wonder how many more dot-com millionaires there might be right now if that had been allowed starting under Reagan. The only difference would be that they couldn't have it till they retired, because it would be in their SS fund.