Nine of the dead were taking part in anti-regime demonstrations after weekly prayers.
"Six people, including a woman, were killed when the security forces opened fire to disperse protests in several neighbourhoods of Homs," in central Syria, a rights activist said.
"Two others were killed in the Qadam neighbourhood of Damascus while one was killed in Daraya" outside Damascus, he said.
State television gave a conflicting toll, saying gunmen killed one demonstrator in Damascus and two people in Homs, one a policeman.
The other two victims, a mother and daughter aged 50 and 20, were killed when the army shelled a chicken hatchery in Al-Bara in northwestern Idlib province, rights activist Bahiya Mardini told AFP in Nicosia by telephone.
She said another family member, also female, was wounded in the shelling in the district of Jabal al-Zawiyah, where soldiers have stormed several villages this week to hunt dissidents.
The army has been using Al-Bara as a base from which to shell neighbouring villages, Mardini reported residents as saying.
Waves of protesters flooded the streets nationwide to demand the fall of the regime, with varying reports putting the turnout in the central city of Hama alone at more than half a million.
The head of the Syrian Observatory for Human Rights, Rami Abdel Rahman, spoke of 500,000 protesters and said this was "the biggest demonstration since the Syrian Revolution broke out" on March 15.
Another activist said that "more than 400,000 marched."
A third said more than 200,000 people gathered in the city's Assi Square, stretching for more than one kilometre, and that there was no sign of security forces.
Hama has a bloody past. In 1982, an estimated 20,000 people were killed when the army put down an Islamist revolt.
An activist said "more than 100,000 people" protested in several districts of Homs as tanks were deployed.
Abdel Karim Rihawi, president of the Syrian League for Human Rights, said "tens of thousands of protesters headed towards Deir Ezzor's Freedom Square upon leaving the mosques" after the weekly prayers in the eastern oil hub on the Euphrates River.
State television reported pro-regime demonstrations in the northern commercial hub of Aleppo and in Suweida, in the south, with people chanting "God, Syria, Bashar and that's it."
Overnight, Abdel Rahman said security forces killed three civilians in Jabal al-Zawiyah, and explosions were heard in the coastal city of Latakia.
Meanwhile Clinton criticised the Assad regime's incoherence in authorising an opposition meeting while stifling dissent.
"Allowing one meeting of the opposition in Damascus is not sufficient," she told reporters on a visit to Lithuania.
"I'm just hurt by recent reports of continuing violence on the border and in Aleppo, where demonstrators have been beaten, attacked with knives by government-organised groups and security forces," Clinton said.
"It is absolutely clear that the Syrian government is running out of time. There isn't any question about that."
Friday's protests followed a call from the Facebook group, The Syrian Revolution 2011 group, which called on people to rally, branding July 1 "the Friday of departure" and saying in a message to Assad: "We don't love you... Go away, you and your party."
Hundreds of protesters in Aleppo, Syria's commercial hub, were beaten back on Thursday by baton-wielding security forces, activists said.
On Thursday, the opposition announced the creation of a "national coordination committee" of exiled dissidents and opponents at home to push for democratic reforms.
The announcement came after about 160 dissidents, several of whom have spent years in jail, gathered on Monday in Damascus, vowing to press ahead with a "peaceful uprising for freedom and democracy and pluralism to establish a democratic state through peaceful means."
The dissidents demanded the right to demonstrate peacefully, the release of political prisoners, freedom of the press, the safe return of refugees and moves to prevent foreign intervention.
The Observatory says 1,360 civilians have been killed since mid-March and that 343 security force personnel have also died. Thousands have been arrested.
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