The protesters had gathered at the local government compound and called on Provincial Gov. Khairullah Batash to step down, the Khaama Press reported.
Clashes occurred between the governor's security guard and the protesters, Provincial Police Chief Mohammad Aslam Begzad said.
Some of the protesters opened fire during the clashes, leading to the deaths and injuries.
The demonstrators have accused Batash and other officials of corruption in a series of protests that began a week ago.