New Zealander David Haussman, one of the detained activists, said group members were trying to contain their excitement.
"At the moment there's a huge sense of relief, for me personally and for my family, but as far as celebrations go, not until I get home to be honest," he told Radio New Zealand this morning.
He added that to be "given amnesty of a crime that we haven't committed is a bit of a strange one".
"There's a lot of steps that have to take place prior to our release, so there's a lot of 'i's to be dotted and 't's to be crossed. To be honest we have no idea how long that may yet take."
Mr Haussman was captured along with fellow Kiwi Jonathan Beauchamp as part of the protest against gas giant Gazprom.
Peter Willcox, captain of the Arctic Sunrise ship, which was boarded and seized in international waters in September, said: "I might soon be going home to my family, but I should never have been charged and jailed in the first place.
"Now it's nearly over and we may soon be truly free, but there's no amnesty for the Arctic."
The amnesty bill has to pass a third reading in parliament before it can come into effect.
However, it is unclear when the non-Russian activists will be allowed to leave the country if the law is passed, or what will happen to the Arctic Sunrise, which remains impounded in Murmansk.
The 30 protesters were released from prison last month on bail, but were not allowed to return home.