A clash erupted between police and a group of masked men who organized the demonstration and then went on to block a main road in the town of Cizre, officials said.
The protesters then stopped vehicles and checked motorists' identifications, police added.
According to police, officers intervened when the men started hurling rocks and Molotov cocktails at them.
Police also said that Celik Salcak, 26, and Umit Kurt, 14, were shot by officers, who responded when some of the demonstrators started firing at them.
Kurt died in the hospital and Salcak was still receiving treatment Tuesday night.
The Turkish government has been active in promoting the "solution process," which aims to secure an end to the decades-long conflict with the PKK.
The PKK is listed as a terrorist organization by Turkey, the U.S. and the EU and the conflict has claimed around 40,000 lives in the past 30 years.
Four people were killed and three others were injured in December 2014 when clashes erupted between pro and anti-PKK groups in the same town, before being broken up by security forces.
In October 2014, protests, which left around 40 people dead, broke out when the Syrian border town of Kobani came under attack by Islamic State of Iraq and the Levant militants and it was alleged that the Turkish government was not doing enough to help the mostly-Kurdish population there.