The following report from the Russian paper Izvestiya notes that Snowden will meet with Russian intelligence, that his flight to Russia was coordinated with the government and WikiLeaks, and that he managed to hoodwink the press who thought he would be on flight to Cuba.

Via Izvestiya (Untranslated can be found here, Google translated below):

Former CIA officer Edward Snowden flew to Moscow with WikiLeaks lawyer Sarah Harrison at full secrecy and for his meeting and the evacuation of the airport was a real special operation carried out with the involvement of foreign diplomats: while distracted the attention of journalists, Ambassador of Ecuador to Russia (his car was parked at the terminal the airport), the whistleblower U.S. intelligence picked up from the plane and taken to the city’s diplomatic machine Venezuelan ambassador. According to experts, Snowden flight to Moscow was agreed with the Russian authorities and intelligence agencies, and despite the fact that his revelations are not a sensation for professionals, representatives of the GRU and the FSB are sure to meet and talk with him.

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U.S. has demanded that Russia expel Snowden, but Russia, like Hong Kong before, is ignoring U.S. requests.
Via WaPo:

Moscow itself has said little to say about Snowden’s presence here. Despite a direct request from the United States to return him to U.S. soil to face charges of leaking government secrets, Russian officials said Monday that they had no legal authority to detain the fugitive former government contractor, who arrived in Moscow from Hong Kong on Sunday and was seeking asylum in Ecuador, reportedly planning to travel there by way of Havana.

A frustrated Secretary of State John F. Kerry said he was troubled by the apparent refusal of fellow world powers China and Russia to respond to espionage charges the United States had filed against Snowden, who leaked top-secret documents about U.S. surveillance programs.

“It is a very serious question for all of us in all our relationships,” Kerry said. “There is no small irony here,” Kerry added, posing the hypothetical question of whether Snowden sought refuge in China and Russia “because they’re such powerful bastions of Internet freedom.”……

Vladimir Lukin, Russia’s human rights ombudsman and a former ambassador to the United States, told the Interfax news agency that Russia had no authority to expel Snowden, as Washington was asking it to do. Russian officials said travelers who never leave a secure transit zone inside an airport — which means not crossing passport control — are not officially on Russian soil. Snowden did not have a Russia visa, several officials said, and therefore could not leave the transit zone.
In addition, Russia and the United States do not have a bilateral extradition treaty, although Kerry said Moscow was obligated to cooperate under international law.

“The Americans can’t demand anything,” Lukin said, referring to the saga dismissively. “Detective stories are good bedtime reading.”

Kerry said the United States has transferred seven Russian prisoners in the past two years and noted: “Reciprocity is pretty important.”“It would be very disappointing if he was willfully allowed to board an airplane,” said Kerry, who was traveling in New Delhi. “There would be without any doubt . . . consequences.”

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There are reports he may leave out on Tuesday, but my guess would be, if he is in the hands of the Russians, they wouldn’t be letting him go until they feel they’ve gotten from him what they want.