The fighting on Tuesday and Wednesday came after Afghan forces claimed to have killed a Taliban commander involved in the kidnapping of 23 South Korean church workers in central Afghanistan in July.
Taliban-led militants are waging a bloody resistance campaign against the Western-supported government of President Hamid Karzai, which replaced the hard-line Islamic militia after the U.S. invasion in 2001.
The two dead soldiers were from NATO's International Security Assistance Force, the alliance said in a statement. It said another ISAF soldier and an interpreter were injured, but gave no more details, including the soldiers' nationalities.
Afghan and coalition soldiers in Shah Wali Kot district, in southern Kandahar province, came under attack while on patrol Tuesday. They fought back before calling in air support, a coalition statement said.
"Surgical and precision airstrikes were carried out on positively identified enemy positions from where machine gun and rocket-propelled grenade fire was originating," the statement said. "Over a dozen insurgents were killed in this engagement."
About 30 kilometers (18 miles) away, insurgents sheltering in a traditional low-walled Afghan compound attacked another joint patrol on Tuesday. Airstrikes later pounded the position, killing six insurgents, the statement said.
In Ghazni province, insurgents early Wednesday attacked a joint coalition and Afghan force, triggering a clash that left "several militants" dead, a coalition statement said. A number of civilians were injured in the clash, the coalition said.
In the southern Helmand province, a remotely controlled bomb exploded under a police vehicle in Gereshk district Wednesday, killing two police and wounding three, said Abdul Manan, a local official.
On Tuesday, Afghan officials claimed to have killed a Taliban commander called Mullah Mateen, who they said was involved the kidnapping of the South Koreans on July 19. The Taliban denied the claim.
Two of the South Koreans were killed soon after the kidnapping, two were released in August, while the remaining 19 were freed last week after Seoul repeated a long-standing commitment to withdraw its 200 troops by year's end, and prevent Christian missionaries from traveling to Afghanistan.
Afghan authorities often make claims they have killed Taliban commanders that turn out to be false. Taliban spokesmen have also downplayed or lied about the extent of their battlefield losses.
The discovery of the submarine has unveiled a few "inconsistencies." For example, how can one explain the fact that the sub was found where it needed to be searched for from the start?
The TurkStream, which runs along the bottom of the Black Sea from Russia's Anapa to Turkey, will consist of two lines, each with a capacity of 15.75 billion cubic meters of gas a year