Prime Minister Gordon Brown has said he "profoundly regrets" the loss of 25 million child benefit records.
He apologised in the Commons for the "inconvenience and worries" caused and said the government was working to prevent the data being used for fraud.
But Conservative leader David Cameron said the government had "failed in its first duty to protect the public".
The Revenue and Customs data on the two missing discs includes names, dates of birth, bank and address details.
During a heated prime minister's questions session, Mr Brown said: "I profoundly regret and apologise for the inconvenience and worries that have been caused to millions of families who receive child benefits.
"When mistakes happen in enforcing procedures, we have a duty to do everything we can to protect the public."
He also said that "the idea that we are complacent about this is quite ridiculous. We are taking all the action that is necessary."
Mr Brown added: "There is no excuse for not following proper procedures."
But Mr Cameron said people would "be angry that the government has failed in its first duty to protect the public."
He added: "What people want from their prime minister on a day like this is to show some broad shoulders, be the big man and accept some responsibility."
MISSING DATA INCLUDES
National insurance number
Name, address and birth date
Names, sex and age of children
Bank/savings account details
Mr Cameron said people were "desperately worried" and they would "find it frankly weird" that Mr Brown still wanted to go ahead with plans for a national ID cards scheme and register.
Earlier, the Tories questioned whether Alistair Darling was "up to the job" of chancellor.
Mr Darling said he "deeply regretted" what had happened, but stressed there was no evidence of misuse of the data.
He told the BBC his confidence had been "shaken" by what he described as a "catastrophic" incident, involving "serious breaches".
"People are entitled to trust the government to look after information that is given to it - for child benefit or any other purposes - and that did not happen here," said the chancellor.
He said the information "should never, ever have left the building in which it was stored".
Mr Darling denied the problem was related to the merger of the Revenue and Customs departments and staff cuts, as claimed by staff members in e-mails to the BBC.
He said banks were monitoring all 7.25 million bank accounts whose details were on the discs, which contained the personal details of all child benefit recipients in the UK.
Mr Darling said anyone who lost money as a result of any misuse of the data would be covered for losses under the banking code.
The entire child benefit database was sent via internal mail by a junior official from HMRC in Washington, Tyne and Wear, to the audit office in London via courier TNT on 18 October.
The chancellor said the civil servant had broken the rules by downloading the data to disc and sending it by unrecorded delivery.
Bosses at the Revenue were not told about what had happened until 8 November and Mr Darling and Mr Brown learned about the situation on 10 November.
The officials involved waited before informing their superiors in the hope that the discs would be found.
Mr Darling reassured those affected that police had no reason to believe they had got into the wrong hands.
People are being urged by both the chancellor and banks to keep a close eye on their accounts "for unusual activity".
The Metropolitan Police is leading the search for the discs, and the Independent Police Complaints Commission, which oversees the HMRC, is investigating the security breach.
Her Majesty's Revenue and Customs has set up a Child Benefit Helpline on 0845 302 1444 for customers who want more details.
LOST CDS - SEQUENCE OF EVENTS
18 October - Junior official from HMRC in Washington, Tyne and Wear, sends two CDs containing password-protected records to audit office in London through courier TNT, neither recorded nor registered
24 October - When package fails to arrive, second one is sent by registered post and arrives safely
3 November - Senior managers are told first package has been lost
10 November - Prime minister and other ministers are informed
12 November - HMRC tell ministers CDs will probably be found
14 November - When HMRC searches fail, Metropolitan Police are called in
15 November- Richard Thomas, Information Commissioner, says remedial action must be taken before public is informed
20 November - HMRC Chairman Paul Gray resigns; Chancellor Alistair Darling makes announcement to House of Commons