The Russians (the aristocracy) hated him. They thought that he was a crazy mystic who had for too much influence on the government. If you look at this contemporary cartoon you see that Rasputin is the puppet master controlling the Tsar and his wife. Also see that the Tsar (on the left) is even smaller than his wife, the implication being that he is being even more controlled than her.

Since it was quite a common belief that the Tsarina was having an affair with Rasputin the reputation of the Tsar was badly damaged. It just looked bad for the emperor of all the Russia's to be cuckolded by some mystic from Siberia.

He was also viewed as uncouth, there are probably fictitious stories of him taking out his penis and waving it around in restaurants. He was kicked out of Kazan cathedral for commandeering the Duma's seats.

Today, your friend is unfortunately correct, to an extent. Most people wouldn't know who Rasputin is, at least right off the bat. I have several Russian acquaintances, two of whom didn't know who he was until I described more details. As for what people think of him if they do know him?

Really depends on who you talk to. Lots of people think it was just some creepy guy who happened to click well with the Tsar's son Alexei and at least appeared to give him some strength (Alexei was a sickly child due to hemophilia). Therefore, the boy's mother adored Rasputin's apparent ability to bring out some life in Alexei. This gave the impression that the Tsaritsa was having an affair (whether or not that was true will likely never be known). On the other hand, I'm sure some people attribute evil/magic powers to him.

That whole ordeal was so long ago, and so removed from recent events, hence why Rasputin is more of a great story to tell and less of an influential historical figure to study. He really didn't do much besides make the royal family look really bad, and influence the Tsaritsa on certain issues.

I'm particularly interested in how people today account for his preternatural charisma and healing abilities.

Official position of Church is that he was a fraud.

Position of most historians is that he was a fraud and possibly a spy.

Is there a prevalent view among common people that something supernatural was afoot?

People who believe in superpowers of course think that he had them. Obviously reasonable people don't.

But he is not popular figure in Russia, people know who he is, like people know who president of France is - but it has 0 relevance in their lives.