The first attack happened at a police checkpoint in Ghazni province, CNN reported. Six people were killed and all the weapons were taken from the checkpoint. A police officer left with the militants who carried out the attack, officials said.
"It means that this attack was a planned trap by the member of [Afghan local police]," said provincial spokesman Fazel Mohammad Sabawoon.
The Taliban immediately claimed responsibility and also said 13 people, not six, were killed at the checkpoint.
In another attack, a suicide bomber claimed the lives of four people in Paktika province, including Ahasanullah Sadat, chief governor of the province's Janikhel district, CNN said.
The Taliban were responsible for that attack, said spokesman Zabiullah Mujahid.