This article has a decent summary of the policy that was under discussion:
The federal government would fund these accounts with annual contributions equal to 2 percent of the wage base used to compute old-age and survivors' benefits under Social Security. Workers’ payroll tax contributions would continue to go into Social Security as before; there would be no “carve-outs” to fund the private accounts. When the worker was ready to retire, the Social Security Administration would calculate a monthly stream of payments based on the balance in her account. If the amount was less than her expected benefit under traditional Social Security, the program would make up the difference. If the amount exceeded her entitlement, she could keep the extra for herself.
It looks like the key difference between this and Bush' 2006 proposal was that the Clinton/Gingrich plan was for an "add on" private account, keeping existing payroll taxes and benefits fully intact as a backstop for those who invested their private account unwisely or unluckily, while Bush's was for a "carve out" account that would redirect some money from the payroll tax system into the accounts and would reduce the benefits formula for those who opted in to the accounts.
If I'm reading it right, the Clinton/Gingrich proposal would also not have increased total payments to retirees unless the returns on their 2% private accounts exceeded their total benefits under the existing system funded by the 12.4% payroll tax (6.2% tax paid by the employee, plus another 6.2% payed by the employer). Otherwise, their benefits would be "topped up" to exactly what they would have gotten without the private account. Which makes the proposal sound less like privatization and more like a way to dress up a two percentage point increase in the payroll tax to make it more politically palatable.