There are several issues about democracy under discussion in Russia. One is corruption and the stories of major public officials, including V. Putin, enjoying lavish palaces--and owning them?--on a government salary. Powerful elected officials after a few years in any country often come to chafe under the limits to personal wealth that coexist with their much less limited public power. That resentment is the seedbed of public pilf in any country, and that seedbed is apparently well-watered in Russia now. The official typically thinks, "Why is it that I can make others rich, but get nothing for myself?" The public thinks, "If you don't like your job, quit!"
But Putin isn't quitting.
In America, presidents are limited to two four year terms, after which they get a reasonably large annual pension and office staff, plus a presidential library named after them. They also can cash in, or not, in the private sector, based on their friendships and name. That seems to suffice. Almost no US presidents are accused of personal enrichment while in office.)
A second issue is whether freedom of speech and freedom of assembly are truly honored in Russia today, or are they offered only as window dressing? In the past, protests were small and could be ridiculed and criticized officially for not following proper procedures for permits, etc. The size of the recent protests make such ridicule ridiculous itself, and thanks, perhaps to calmer voices in the Kremlin, the approach of mockery has been muted.
The existence of free speech and freedom of assembly actually may be honored more now by the Putin regime than in recent years. Evidence is the way the whacky assertion of Mr. Putin that Hillary Clinton had inspired the protest demonstrations was laughed down. There even were nightclub routines making fun of it, and finally President Medvedev allowed that the protests are home grown. Also lending credibility to the protests, one sees the remarkable story that the Russian Orthodox Church now feels confident enough to praise the protesters. Anywhere in the West, that would hardly be news (churches love protests of government), but it's a novelty in Russia and, paradoxically, suggests a liberalizing of the regime. After all, in a fully controlled society the Church wouldn't dare raise its voice.
In short, the recent poor showing of Mr. Putin's United Russia party and the way the prime minister has been booed on occasion mark a humbling of a leader who is known for his arrogance. On the other hand, a humbler Putin might be just the ticket for electoral success next March. The protestors are only a small slice of a vast population and polls still show that Unite Russia, headed by Putin, is likely to win the coming Presidential elections. It's just that the Kremlin is being reminded that in a democracy the people are sovereign, and you can alienate them.
But that still leaves issue number three: the recent apparent electoral fraud. There have been a few firings and promises of investigations. But a truly confident regime would conduct real investigations, come to honest conclusions and admit fault as appropriate. That might do more to gain credibility for the regime than anything they could do right now.
Don't count on it anytime soon.