The government issued new bonds, which spread some of the cost over the generations. Privatization of state-owned business, and a reduction in government spending elsewhere, were also important. In the meantime, during the transition, everyone contributing to the old system could remain in it, but those who moved had their rights to partially accrued pension income guaranteed by the government. All new entrants to the work force were required to go into the PSA system. But the costs are low - and are dwarfed by the higher returns the PSA system generates.
Private pensions are less reliable and safe.
In fact, it's hard to consider the present setup reliable, with governments increasing taxes and decreasing payouts. The investment results of private funds cannot be guaranteed. The government supervises the investment companies, and of course the fund managers themselves keep a constant watchful eye on the accounts. The PSA system has other benefits. For example, if this system were adopted Europe-wide, workers would not risk losing their pension rights if they left a job in one country for a job in another.