The Prime Minister came under fire for cuts to public services and her plans for social care, while Mr Corbyn faced questioning on his attitude to security issues and past comments about the IRA and the Falklands War.
Afterwards, both sides claimed victory, with Labour saying Mrs May had "floundered" on public services, while the Tories said Mr Corbyn's comments on security were "deeply worrying".
Appearing on a Sky News/Channel 4 "Battle for Number 10" broadcast, the Labour leader refused to be drawn on whether he would authorise a drone strike against a terrorist plotting overseas to attack the UK.
"It is a hypothetical question," he said. "We have to look at the evidence that is there at the time to make that fatal decision one way or the other."
Mr Corbyn, a lifelong opponent of nuclear weapons who has made clear that he would never authorise their use, nevertheless indicated he would issue the customary final instructions to the commanders of the Trident submarine fleet if he became prime minister.
During her session from the studio audience, Mrs May was accused by a police officer of presiding over "devastating" cuts, asked by a midwife to justify her "chronic underfunding" of the NHS and heckled over school funding.
The Prime Minister insisted that she was determined to take on difficult issues and do the right thing for the country.
"If, in order to address them and do the right thing by the country, it takes being a difficult woman, then that's exactly what I will be," she said.
Mrs May also reaffirmed that she would walk away from the forthcoming Brexit negotiations without a deal rather than accept a "bad" deal.
"You're not in there to get a deal at any price," she said.
Under the format agreed by the two parties, the two leaders each faced 20 minutes of questions from the audience followed by 18 minutes in front of Mr Paxman, with Mr Corbyn going first.
One of the first questions to Mr Corbyn came from an audience member who claimed the Labour leader had "openly supported the IRA in the past" by attending a commemoration for eight IRA members killed by the SAS in Loughgall in 1987.
Mr Corbyn said: "The contribution I made to that meeting was to call for a peace and dialogue process in Northern Ireland."
He was pressed by Mr Paxman over comments he made following the Argentine invasion of the Falklands that "young unemployed men" were being sent to the South Atlantic to die in pursuit of a "Tory plot".
The Labour leader said he did not believe it had been a "plot" but added: "Margaret Thatcher made a great deal of that issue at the time. I felt that she was exploiting the situation."
Mr Corbyn defended his description of the Palestinian group Hamas as "friends" and his comment that the killing of Osama bin Laden by US special forces had been a "tragedy".
"I think he should have been arrested and he should have been put on trial. And he could have been," he said.
Challenged by one man who said he liked the Labour manifesto but did not regard Mr Corbyn as "someone who could run this country", the Labour leader said he saw himself as a listening politician.
"You should never be so high and mighty you can't listen to somebody else and learn something from them," he said.
During her session in front of the studio audience, Mrs May insisted she had given the police the resources they needed but acknowledged that numbers in England and Wales had fallen by around 20,000.
"What we had to do when we came into government in 2010 was to ensure that we were living within our means and that was very important because of the economic situation we had inherited," she said.
She was pressed by Mr Paxman over her climbdowns on a proposed Budget hike in national insurance and her social care changes just days after they were unveiled in the Tories' election manifesto - and what it would mean for the Brexit negotiations.
"What one's bound to say is that if I was sitting in Brussels and I was looking at you as the person I had to negotiate with, I'd think, 'She's a blowhard who collapses at the first sign of gunfire'," he said.
She replied that as home secretary she had record of negotiating with Brussels and "delivering for this country on a number of issues on justice and home affairs which people said we were never going to get, and I got those negotiations".
A spokesman for Mr Corbyn said: "Theresa May floundered on her record on police cuts, on funding for our NHS and schools and on her manifesto policy on social care that didn't last more than a few days before it was amended with an unspecified cap."