Monday’s violence was the latest in a series of fatal clashes in the region where two decades of rebellion against Indian rule have left thousands dead.
Eight Kashmiri civilians have been killed in incidents involving Indian security forces in less than three weeks, with each death sparking further violent protests leading to more firing by paramilitary troops.
Monday's deaths sparked spontaneous shutdowns in Srinagar and other towns of the Kashmir valley, witnesses said. Residents poured on to the streets chanting "We want freedom!" and "Allah is great!"
In the ensuing clashes, some 40 protesters and 25 policemen were injured across the valley, police said.
Sopore has been under curfew since Friday after two young men died when soldiers opened fire as protesters attacked their vehicle.
Authorities closed all schools and colleges in the Kashmir valley for two days in an attempt to thwart the unrest after separatist groups called on students to hold protest rallies against Indian rule.
In an earlier incident on Monday, seven people were injured when security forces clashed with protesters in Sopore. The clashes erupted when mourners tried to carry the body of a man killed on Sunday through the streets and security forces fired rubber bullets and tear gas to disperse the crowds.
Police said security forces had acted to control stone-throwing demonstrators.
India maintains a massive troop presence in Kashmir to battle separatist protesters.
Nuclear-armed rivals India and Pakistan each control part of Kashmir but lay claim to all of the Himalayan region.