Snowden has avoided all contact with media since arriving in Russia on a flight from Hong Kong in June and his lawyer Anatoly Kucherena has become his unofficial spokesman.
"I am his only link with the outside world at the moment. Even his contacts with his parents are carried out through me," Kucherena said in an interview published in Itogi weekly magazine.
"He would walk past you and you wouldn't recognise him," he told Itogi.
"It's a question of clothes and small alternations to his appearance. So I'm not deceiving anyone: he really does walk freely around on the streets."
Snowden has also made quick progress in learning Russian, his lawyer said.
"He is an extremely fast learner as far as the Russian language is concerned," Kucherena said in another interview that will air today on Kremlin-funded RT television.
"He only needs a few hours or days to learn the ropes and start speaking," the lawyer said in comments dubbed into English.
The 30-year-old former CIA security analyst is wanted by the United States after revealing details of massive surveillance by the National Security Agency to the media.
While describing an isolated existence, Kucherena said Snowden did not regret his decision to give up his life in the United States.
"He's not disappointed. He believes he did everything right," he told RT.
Kucherena, a high-profile lawyer who is an advisor to President Vladimir Putin, said he is working for free, as Snowden's personal money is running low.
"I can't give you certain dates but soon he will come to Russia and meet with his son. There will be him and his mother and probably one of his grandparents."
He warned however that a family visit could threaten Snowden's safety, since US intelligence could use it to find his hideout.