The most recent proposal to privatize Social Security was put forward by Pres. Bush in 2004 during his campaign for reelection. After being reelected, he put some effort into pushing his plan, but was unable to get it passed by Congress.
Bush's plan was, basically speaking, to allow people to pay into private accounts instead of Social Security. In such a plan, a person's savings would actually belong to them rather than going into the general Social Security pool. This money would have been invested in one of a variety of plans (various mixes of stocks and such). The person who put the money in would get back whatever they put in plus however much money it made through dividends and having the prices of stocks and such go up. This would have been in contrast to the current system where all the money that people put into Social Security goes into a big pool that is paid out to retirees based on formulas set up by the government.
It is hard to know whether this would have been good or bad for investors. In the aftermath of the financial crash of 2008, it does not look good because of how much the stock market fell. This kind of volatility in the stock market scares people and makes them want to have the more "guaranteed" return of Social Security rather than gambling on stocks.