Elections had been due to take place by the end of 2017 under a church-mediated deal but were further delayed, and the poll is now scheduled for December 23, 2018.
"Eleven people died in Kinshasa and one in Kananga", Jonas Tshombela, a spokesman for the protest organisers, told AFP.
Florence Marchal, spokesperson for the United Nations peacekeeping mission in Congo, said security forces had shot dead at least seven people in Kinshasa.
Police spokesman colonel Rombaut-Pierrot Mwanamputu, meanwhile, said in a statement on Tuesday that "no deaths" had occurred in the context of the demonstrations.
More than 100 protesters have been detained so far by the police.
A government statement said one policeman had also been killed. "What's more, you have a white man with you - that's a race that causes us problems".
In Kinshasa, Catholics of the "Lay Coordinating Committee" had invited worshippers to walk, holding bibles, rosaries and crucifixes, after mass on Sunday.
One parishioner who identified herself as Chantal said: "People fell, first-aiders are resuscitating old ladies who have fallen" - but added the priest carried on saying mass.
Internet and SMS services in the Democratic Republic of Congo have been restored days after the government ordered service providers to cut them ahead of planned anti-government protests.
He urged all Congolese political actors to remain committed to the December 2016 political agreement paving way for elections possibly before the end of this year.
Despite pressure from overseas, the authorities seem intent on pursuing a clampdown, flooding cities with police and troops whenever the opposition tries to make a show of strength, analysts said.
"The major deployment of police, army and military gear aims at discouraging people who are tempted to go out and demonstrate", said Congolese analyst Jacques Wondo.
Vital Kamerhe, head of the third biggest opposition party in parliament, argued that the presence of priests in Sunday's demonstrations showed that the protest movement had gone into "higher gear".
"We tell ourselves we are in a democratic state, but we are not in a democracy".
At the Notre Dame cathedral in the northern Lingwala district of Kinshasa, security forces deployed tear gas as opposition leader Felix Tshisekedi arrived, AFP reporters said.