The fighting for control of Sirte, hometown of slain dictator Moamer Kadhafi, has been raging since Tuesday, with one top Libyan diplomat warning of a "massacre" in the city.
LANA said the 12 people beheaded were local gunmen who had been battling IS in the eastern Sirte district known as "neighbourhood three".
The agency also reported that IS militants executed 22 other Sirte residents who had taken up arms against the jihadist group as they lay wounded in a city hospital.
A Sirte city council official told AFP that clashes continued to rage on Saturday.
"There is continuous fighting, especially in neighbourhood three, and the casualty toll is rising," said the unnamed official.
Libya's ambassador to France, Chibani Abuhamoud, told AFP on Friday that fighting in Sirte had left between 150 and 200 dead.
"A real massacre is taking place, and we call on the international community to intervene," the diplomat said.
The ambassador answers to Libya's internationally recognised government, which has taken refuge in the east of the country and is in a power struggle with a rival administration based in Tripoli.
Media loyal to the Tripoli authorities said warplanes were bombing armed groups linked to IS in Sirte.
The Libyan ambassador said the fighting erupted after IS assassinated earlier this week an influential imam from the powerful Al-Farjan tribe.
The jihadists have since been "massacring people, even killing people in their homes," Abuhamoud charged.
Late Friday, the Dar al-Iftaa Muslim organisation that issues religious decrees issued a statement calling on Libyans to mobilise against IS.
Libya has descended into chaos since the fall of Kadhafi in 2011.