Government censorship of the Internet is a cat-and-mouse game. And despite more aggressive tactics in recent months, the cats have been largely frustrated while the mice wriggle away.

But this year, the challenges for Silicon Valley will mount, with Russia and Turkey in particular trying to tighten controls on foreign-based Internet companies. Major American companies like Facebook, Twitter and Google are increasingly being put in the tricky position of figuring out which laws and orders to comply with around the world — and which to ignore or contest.

On Wednesday, Russia’s president, Vladimir V. Putin, signed the latest version of a personal data law that will require companies to store data about Russian users on computers inside the country, where it will be easier for the government to get access to it. With few companies expected to comply with the law, which goes into effect Sept. 1, a confrontation may well erupt.

The clumsiness of current censorship efforts was apparent in mid-December, when Russia’s Internet regulator demanded that Facebook remove a page that was promoting an anti-government rally. After Facebook blocked the page for its 10 million or so Russian users, dozens of copycat pages popped up and the word spread on other social networks like Twitter. That created even more publicity for the planned Jan. 15 event, intended to protest the sentencing of Aleksei A. Navalny, a leading opposition figure.

Anton Nosik, a prominent Russian blogger whose work has been censored by regulators, said it was absurd for a government to think it could easily stamp out an article or video when it can be copied or found elsewhere with a few clicks. “The reader wants to see what he was prevented from seeing,” Mr. Nosik said in an interview. “All that blocking doesn’t work.”

Instead, that prompted the government to switch tactics, moving Mr. Navalny’s sentencing to Dec. 30 with little notice in an attempt to diminish protests.

The Turkish government faced similar embarrassment when it tried to stop the dissemination of leaked documents and audio recordings on Twitter in March. The administration of Recep Tayyip Erdogan, who was then prime minister and is now president, ordered the shutdown of Twitter within Turkey after the company refused to block the posts, which implicated government officials in a corruption investigation.

Not only did the government lose a court fight on the issue, but while Twitter was blocked, legions of Turkish users taught one another technical tricks to evade the ban, even spray-painting the instructions on the walls of buildings.

“We all became hackers,” Asli Tunc, a professor of communication at Istanbul Bilgi University, said in a phone interview. “And we all got on Twitter.”

Despite such victories for free-speech advocates, governments around the world are stepping up their efforts to control the Internet, escalating the confrontation.

“The trendlines are consistent,” Colin Crowell, Twitter’s global vice president of public policy, said in a phone interview. “There are more and more requests for removal of information.”

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Pakistan, for example, bombarded Facebook with nearly 1,800 requests to take down content in the first half of 2014, according to the company’s most recent transparency report. Google’s YouTube video service has long been blocked there. And the government briefly succeeding in getting Twitter to block certain “blasphemous” or “unethical” tweets last year until the company re-examined Pakistani law and determined the requests didn’t meet legal requirements.

It’s not just autocratic regimes that are pressing for limits on free speech. In the European Union, a court ruling last year established a “right to be forgotten,” allowing residents to ask search engines like Google to remove links to negative material about them. Now privacy regulators want Google to also delete the links from search results on the non-European versions of its service because anyone in Europe can easily get access to the alternate sites.

Free-speech activists view Facebook, the world’s largest social network with 1.35 billion monthly users, as the company most inclined to work with governments and do whatever is necessary to keep its service up and running.

Aleksei Navalny broke house arrest to attend a rally in his support. Credit Anton Belitskiy/Reuters

Last spring, while Twitter was blocked in Turkey and YouTube was shut down, Facebook removed contested content and continued to operate. It has a dedicated team of outside lawyers who field censorship requests from the Turkish government and then recommend to corporate officials whether content should be blocked.

“Facebook can be quite important to the people who use it, so we try to make sure it remains accessible,” a company spokesman said. “We aggressively push back on unlawful or overly broad government requests.”

Twitter, which has about 284 million monthly users, styles itself as the world’s town square and a global champion of free speech, conforming to the letter of censorship laws while winking at workaround strategies, like users changing the location listed on their profile to evade specific blocks that apply in a particular country.

For Turkey’s opposition movement, Professor Tunc said, Twitter “basically created an opening, a refreshing alternative, especially during the protests. And they know that. They act like a defender of freedom.”

As the biggest player, Google, whose YouTube service seems to draw the particular ire of foreign governments, has been forced into fights on many fronts. It is still viewed by many as a hero for its decision to pull out of China in 2010 rather than continue to censor search results there.

The company explained its philosophy at that time: “We have a bias in favor of people’s right to free expression. We are driven by a belief that more information means more choice, more freedom and ultimately more power for the individual.”

While China remains a thorn in the side of most Western Internet companies — Facebook and Twitter are basically blocked there — Russia is the current flash point in the censorship wars.

Over the summer, the Russian government began demanding that anyone with at least 3,000 daily visitors follow rules similar to those applying to a media company and face content restrictions. So far, Twitter and Facebook are simply passing those requests along to their users without making sure anyone complies. Many do not, but so far the Russian government has not pressed the issue.