Gen Zelmia Oryakhail, provincial police chief of Paktia province, said the bomber was on a motorcycle and detonated his explosives at midday outside the market in Samkani district as American forces passed.
He said a local school had just let pupils out for lunch.
The US military coalition in Afghanistan confirmed that two of its service members died in the explosion. It did not confirm their nationalities.
Ten schoolchildren and one Afghan police officer were killed, the Afghan Ministry of Interior said. It did not say the ages of the children.
The insurgents have also littered the country's roads with homemade landmines and roadside bombs that often kill civilians.
Also today, a landmine killed seven Afghan civilians in the eastern province of Laghman. A statement from the provincial government said a group of four women and two children had gone with a male driver into the hills to collect firewood. On their way back, their vehicle hit the mine and all inside were killed.
As the withdrawal of most international forces looms, there are tentative parallel efforts towards negotiations.
The Taliban confirmed today that it sent a delegation to Iran for three days of talks, signaling that Tehran could be seeking the role of regional mediator in negotiations to end its neighbour's 12-year war.
