A new "arms-length" watchdog is needed to oversee the UK's security services, a group of MPs and peers has said.
MI5 employs about 1,900 people
Parliament's joint committee on human rights says there is an "increasingly urgent need" for more accountability.
The committee has also criticised MI5 chief Eliza Manningham-Buller for not meeting it over claims the CIA flies prisoners around the world via the UK.
MI5, MI6 and GCHQ currently answer to Parliament's intelligence and security committee (ISC), appointed by the PM.
"There is an increasingly urgent need to devise new mechanisms of independent accountability and oversight of both the security and intelligence agencies and the government's claims based on intelligence information," the human rights committee's report said.
Its chairman, Andrew Dismore, said: "It's not a question of revealing state secrets, but it's a question of being able to reassure the public through an arms length body, that the security services are being properly scrutinised."
Dame Eliza says she does meet another committee which operates privately.
The report points to Canada as an example where there is a review committee which keeps watch on the performance of the security agencies and investigates complaints.
The call comes as the families of those killed in last year's London bombings continue to demand a public inquiry - a request the government has so far refused to grant.
Eliza Manningham-Buller declined to be questioned
Public confidence in the security and intelligence services also fell when weapons of mass destruction were not found in Iraq.
The human rights committee has unusually published its correspondence with Dame Eliza, asking her to appear before the committee.
Members wanted to ask her about the steps MI6 takes to ensure it knows whether information it receives from overseas agencies was obtained by torture.
And it wants to ask about "extraordinary renditions" - the alleged flights carrying suspects secretly around the globe.
In her response, Dame Eliza said it would be inappropriate to discuss government policy and that she could give the ISC sensitive information in a way it could be protected.
But in its report, the committee said: "We regret that we did not have the opportunity to ask her a number of important questions of concern to us in connection with this inquiry.
"We have no desire to obtain access to state secrets, but we do consider it a matter of some importance that the head of the security services be prepared to answer questions from the parliamentary committee with responsibility for human rights."
The MPs and peers also call for an end to the ban on using phone-tap evidence in British courts.
There are enough possible safeguards, including using security-vetted lawyers, to reassure the intelligence agencies that their sources would not be compromised by the move, they say.
And they want new incentives for people to give evidence to prosecute suspected terrorists, including witness protection schemes and lower sentences for "super grasses".
The committee does not want the kind of continental-style system where judges and magistrates lead investigations.
"We are firmly of the view that the investigating magistrates model should not be borrowed wholesale and imported into our own institutional arrangements," it said.
"Nor do we think that there is anything in the investigative approach which might be borrowed or grafted on to our more adversarial, common law tradition."
Dame Pauline Neville-Jones, who worked for MI5 and chaired the joint intelligence committee, told the BBC she had sympathy with the idea there should be more accountability, but was unconvinced a separate body was the answer.
She said one way forward would be for a member of the opposition to chair the security committee, in the same way that Tory MP Edward Leigh chairs the public accounts committee.
"We do need to have some confidence that the committee is independently minded. It could have rather more open and fuller reports," she said.