It seems to be a complicated story, but the Hungarian (actually Orban's) government appears to have been undermining the private pension system for years, leading to a loss of confidence in [whatever was left of] it. According to Reuters
The government has already been tapping the private schemes since 2010 for 360 billion forints ($1.5 billion) a year to cut the state budget deficit. [...]
Mandatory private funds, running alongside the state pension system and elective private pension funds, were central to a 1997 scheme to reduce state spending.
But Orban’s government went on to nationalise more than 90 percent of their assets, worth about $12 billion. Members could declare an intent to keep their savings with the funds, but few did so.
According to sources closer to position of Hungarian government, the Hungarian private pension system was plagued by high overhead and poor performance (even before the 2008 financial crisis, when it took a significant hit):
As the result of a 2009 study, OECD described the high cost level of the Hungarian mandatory private pension funds system as a negative example (OECD, 2009). Of all mandatory pension funds systems, members had to pay the highest administrative and management costs in Hungary, amounting to around 2% of the wealth managed by the system, while the fees were the lowest in Sweden (less than 0.5% of the managed wealth).
Mostly as a consequence of the above described actions, the return of Hungarian mandatory private pension funds was especially low in the 13-year-long period between 1997 and 2010 and did not even on par with ination; therefore, it had a negative return. None of the mandatory private pension funds operating in Hungary reached the level the index of short-term government bonds (RMAX), but they were signicantly lower. Consequently, Hungarian mandatory private pension fund members would have been better o if these funds had invested the savings of their members into Hungarian government bonds (Magyarorszag Kormanya, 2012, p. 6) [...]
By the end of 2010, Hungary faced a dilemma due to the obligation of the EU’s mandatory targeted budget deficit of 3.8% by 2010 and 3% by 2011 and the constantly increasing deficit caused by private pension funds. Hungary had to choose whether to eliminate and nationalise (similarly to Argentina) second-pillar of pension system or to follow IMF’s recipe and the practice of the preceding government and further reduce pensions. However, the Hungarian government chose a third solution by not nationalising the private pension funds, but removing the protective screens of the state from them. As a result, the government does not guarantee private pension funds that their members will benefit even if their private pension funds make unfavourable financial decisions. Private pension funds still had the opportunity to provide above average pensions to their clients, but they lost all state-guarantee and had to fight real market risks. Also, this change made it possible to stop the decrease of the real value of pension in the case of those in the state pillar.
The World Bank, which is critical of the nationalization move, nonetheless acknowledged many of the problems with Hungary's 2nd pillar, in particular (I'm selecting a few points, they have considerably more):
The badly designed institutional structure led to low accountability of financial providers.
Fund managers broke even after appr. 4 years.
Regulatory changes (fee caps, introduction of life-cycle investments) came only late, with unlucky timing.
Legal changes regarding ownership of funds and of pay-out design could not be enacted in 2010. [Since from the 1998 start:] Financial providers, asset managers do not own the fund and do not have to put up real capital.
The market went for a low-risk home-biased herding strategy for a long time.
The overall result is an average net real return of about 1% in 1998-2010.
The effect on public confidence of those 2010 quasi-nationalization measures appears to have been substantial:
The decision by the Hungarian government was highly controversial and justified by the need to cut the budget deficit. However, the controversy was fuelled by the government’s announcement at the eleventh hour that the suspension would be permanent and the second pillar would be removed altogether, says Daniel Gera, an associate in the employment team at French legal firm Gide Loyrette Norel’s Budapest office.
Consumers were given the option to transfer their account into the existing private sector third pillar, he says, but this was made difficult and the deadline was very short. And there was worse to come.
“If you wanted to keep a private account, there was a risk you would not receive your state pensions entitlements accrued to that date. Although this was ultimately declared unconstitutional, by then only 3 per cent remained outside the first pillar,” says Gera.
The impact of this measure was far reaching. It effectively ended the private provision of pensions in Hungary and greatly knocked public confidence.
As far as Romania goes, their 2nd pillar was introduced in 2007 (compared to Hungary in 1998), so perhaps it was better designed and/or there was less of a piggy bank [for the state] to raid... Also, the Orban government was eager not to use much money that IMF made available to them during the financial crisis. Romania did however cut the contribution to the 2nd pillar, recently, in 2017 from 5.1% to 3.75%. And in 2018 they capped the 2nd pillar administration fee to 1%, from 2.5% previously. Also in Romania as of that date:
Savers will be able to take their money out at a fee of 2 per cent of their assets, in a blow to the country’s second pillar of private pension schemes.