The Washington Post has a human-readable explanation of the Bush privatization plan Social Security. The article explains, I think handily, how the private fund would work, and by what amount of the gov't portion would be reduced. As right wingers are quick to point out, the private accounts are transferrable upon death, unlike the gov't pension, and thus represent an asset.
However, two factors need to be taken into account, the first from the same WaPo article, namely, the Congressional Budget Office calculates that, adjusting for risk, and including overhead costs, the new accounts will result in zero net benefit for the citizen.
The second, which I derived myself, says this. If there is no net benefit change and the money is converted to relatives upon death, then the system will have even higher deficits than currently for no benefit. Although, to be fair, the slightly increased deficits are part of the transfer from young diers to old, which is often cited as a weakness in the system.