Hillary Clinton has called for a "sensible adult conversation", to be held in a transparent way, about the boundaries of state surveillance highlighted by the leaking of secret NSA files by the whistleblower Edward Snowden.

In a boost to Nick Clegg, the British deputy prime minister, who is planning to start conversations within government about the oversight of Britain's intelligence agencies, the former US secretary of state said it would be wrong to shut down a debate.

Clinton, who is seen as a frontrunner for the 2016 US presidential election, said at Chatham House in London: "This is a very important question. On the intelligence issue, we are democracies thank goodness, both the US and the UK.

"We need to have a sensible adult conversation about what is necessary to be done, and how to do it, in a way that is as transparent as it can be, with as much oversight and citizens' understanding as there can be."

Her words were echoed by the British shadow home secretary, Yvette Cooper, who repeated her call in a speech in July for reform of the oversight of the intelligence agencies. Cooper, a former member of the parliamentary intelligence and security committee that oversees the agencies,said: "I have long argued that checks and balances need to be stronger – this would benefit and maintain confidence in the vital work of our security and intelligence agencies as well as being in the interests of democracy."

The conciliatory language of Clinton and Cooper contrasted with that of MI5, whose director general, Andrew Parker, warned earlier this week that the leaked documents by Snowden had provided a gift to terrorists.

The former Labour foreign secretary Jack Straw reinforced that message Friday, criticising the Guardian for publishing articles based on the leaked documents.

Straw, foreign secretary during the Iraq war in 2003, told the BBC: "They're blinding themselves about the consequence and also showing an extraordinary naivety and arrogance in implying that they are in a position to judge whether or not particular secrets which they have published are not likely to damage the national interest, and they're not in any position at all to do that."

Clegg, who agrees with Straw that in some cases the Guardian was wrong to publish details from the NSA files, believes the leaks show the need to consider updating the legal oversight of Britain's security services. Aides said he would be calling in experts from inside and outside Whitehall amid concerns that the leaked files show that powerful new technologies appear to have outstripped the current system of legislative and political oversight.

Vince Cable, the business secretary, confirmed the Lib Dems wanted to examine the oversight of the intelligence agencies and he praised the Guardian for performing a public service in publishing articles on the files.

He told BBC Radio 4's Today: "I think the Guardian has done a very considerable public service. Snowden's contribution is two-fold. One is a positive one, which is whistleblowing, and the other is more worrying, that a large amount of genuinely important intelligence material does seem to have been passed across.

"The conclusion which Nick Clegg came to, and set out this morning, is that we do need to have proper political oversight of the intelligence services and arguably we haven't until now. What they [the Guardian] did was, as journalists, entirely correct and right. Snowden is a different kettle of fish."

Downing Street indicated that senior members of the coalition were at odds when the prime minister's spokesman dismissed Cable's claim that Britain arguably lacks a proper system of oversight, saying that the prime minister is satisfied with the system. But David Cameron's spokesman added that members of the national security council, of which Clegg is a member, were entitled to question the intelligence agencies.

The spokesman said: "There is a debate that is outside of government that is often reported in [the Guardian] and other newspapers. There is the scope for members of the national security council, privy councillors, to ask questions and the like to better understand the work that the agencies do. That is always open to them."

The agencies are overseen in three ways in Britain: they are answerable to their relevant secretary of state; accountable to parliament's intelligence and security committee chaired by the former Tory foreign secretary Sir Malcolm Rifkind; and answerable to the intelligence commissioners.

David Bickford, a former legal director of MI5 and MI6, told the Guardian that the current oversight regime for Britain's intelligence agencies was "obviously inadequate."

"Secrecy in this country is over-protected and under-regulated," he said. "The UK has signally failed to prepare itself for openness when dealing with politically sensitive issues such as terrorism or the involvement of their secret agencies in the gathering of information by secret means."Bickford added: "We see only a fleeting and ephemeral face of the intelligence agencies chiefs; ministers glide over the threats, never explain their relationship with those agencies and are content to retain an obviously inadequate system for their supervision."

Bickford said public scepticism was "made worse by the Communications Data Bill's proposal that the agencies themselves control their mining of communications data."

He added: "Unless government takes this debate seriously, secrecy will be pierced by the needs of society and terrorism and organised crime will plunder our sovereignty."

Nick Pickles, the director of Big Brother Watch, joined forces with ten other like-minded campaigners to call on Cameron and Clegg to reform the system of oversight of the intelligence agencies. In a letter to the prime minister, deputy prime ministers and the ISC chairman the campaigners called for an independent review of the Regulation of Investigative Powers Act 2000 and the Intelligence Services Act 1994; the publication, in line with the practice in the US of legal opinions used to support surveillance methods; and to allow the Intelligence and Security Committee to report directly to Parliament rather than just to the prime minister.

In the letter they said: "We would be delighted to meet with you or members of your Government to discuss these issues. At a time when the internet is an inescapable part of daily life, the modern economy and the delivery of public services, it is surely paramount that the laws that govern surveillance are fit for a digital age, and that the safeguards that operate are robust, properly resourced and can command public confidence."

Sir Francis Richards, a former GCHQ director, questioned whether it was right for an MP of a governing party to chair the intelligence and security committee. Richards, who was highly critical of the Guardian for publishing the leaked documents, told The World at One on Radio 4: "I think it's probably not a very good idea that a former senior minister in a Conservative government is the current chair of the intelligence and security committee."

In her remarks, Clinton did not comment on the UK's oversight arrangements. But she indicated she was wholly supportive of the approach adopted by Barack Obama who – in contrast to Downing Street – has said he welcomes a debate on surveillance in the wake of the NSA leaks.

Answering a question from the Guardian at Chatham House, she said the discussion had to take place within a framework that addressed issues of privacy and protection of citizens because some surveillance programmes remained a "really critical ingredient in our homeland security."

Clinton, who is considering whether to make her second challenge for the Democratic presidential nomination, added: "It would be going down a wrong path if we were to reject the importance of the debate, and the kinds of intelligence activities that genuinely keep us safe.

"So how do we sort all of this out? This is a problem that is well over a decade old, where these capacities have corresponded with increasing outreach to consumers on the business side and increasing concern about security on the government side. People need to be better informed."

Cooper was careful to praise the work of the intelligence agencies. She said: "The work of the security and intelligence services is vital, and most of it by necessity must be kept secret to protect our national security and public safety. Leaks of classified information can be deeply damaging, and in the wrong hands can place our country and people's lives in danger. But because so much information needs to be kept secret, it's even more important that there are strong checks and balances in place, to provide effective oversight, accountability and reassurance to the public. And that's why we are proposing reforms.

"When dealing with international terrorism or other complex and serious threats, strong powers will sometimes be needed but they should be matched by strong checks and balances to ensure power is not concentrated or abused, and to investigate when things go wrong.

As a former member of the Intelligence and Security Committee I have long called for it to have greater powers and resources to pursue searching investigations. And it also needs the credibility and independence of government in the eyes of the public, Parliament and the media that the Public Accounts Committee has.

"We should be strengthening other forms of oversight too. We are way past the time when oversight can be delivered simply by retired judges undertaking paper-based reviews and never speaking publicly about their conclusions or the work they do. The current framework of commissioners who few people have heard of simply isn't adequate for a world of rapidly changing technology and public expectations.

"The very nature of intelligence agencies is that their work needs to go on behind closed doors, and many of the checks and balances need to be behind closed doors too. But those checks and balances need to be strong enough and credible enough so the public can be confidence in the vital work the agencies do."