The Russian billionaire and former KGB spy bought The Independent and The Independent on Sunday for the sum of £1 yesterday.
This follows Lebedev's purchase of the Evening Standard for £1 in January.
Unity co-founder Gerry Hopkinson said: ‘The challenge for Lebedev is to allow the editorial team to get on with their jobs. I don't want to damn the man but I guess I hope they carry on in the tradition of [former owner] Tony O'Reilly.
‘Lebedev has kept the Evening Standard in good shape - so far so good. So if we get a good story I'd hope we'd get a good hearing - I don't think that's going to change.'
Independent News & Media (INM), the Dublin-based owner of the titles, agreed to pay Mr Lebedev £9.25m, to cover the £1m a month the papers are expected to lose this year.
MD of Grayling's Moscow office Stephen Lock said: 'When it comes to Russia, one can drive oneself crazy trying to see the games within the games being played. But as a long-term move, having a solid media block in the UK which, one day, may write more objectively about Russia, is good for Lebedev both as a businessman within Russia and as an opinion former about Russia's future. Russians, remember, play a very long game which they can mask very well. These are some very nice chess pieces on Lebedev's board.'