A mass rally in Moscow and similar demonstrations in two other cities were called after parliament backed the controversial bill last month.
The government says the bill, which allows it to isolate Russia's internet service from the rest of the world, will improve cyber-security.
But campaigners say it is an attempt to increase censorship and stifle dissent.
Activists say more than 15,000 people gathered in Moscow on Sunday, which is double the estimate given by the police.
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