It's hard to see how the answer to this question isn't simply "yes." Social media makes it easier to communicate to a larger group of people at significantly cheaper cost, from anywhere in the world. That's true of both private and state actors, and of good actors and bad ones.
If the USSR wanted to message to Western audiences, their abilities were quite limited. Nearly everyone consumed their news through TV or newspapers, which served as gatekeepers for the kind of messaging they allowed. Overt pro-Soviet messaging would be fact-checked, interviews would be confrontational, and infiltration was a costly, risky business. There were papers and magazine put out by local Communist parties (ostensibly independent, actually controlled by the Soviets), but they never had much reach outside of the already converted.
More recently, when cable became a mainstay and anyone could pay companies to host their channel, Russia tried their hand at an overtly state-owned outlet called Russia Today. When it comes to TV, it was a dismal failure:
RT, the documents note, is not present in Nielsen ratings for the U.S.
for 2012, which it says start with channels with an audience of 18
million households. Nor does it make cable news channels rankings,
meaning that, according to the documents, “the average daily
viewership of RT programs in the US does not reach [30,000] people.”
The documents say that its viewership doesn’t even amount to 0.1
percent of Europe’s television audience, except in Britain, where
the 2013 viewership was put at about 120,000 people daily: “In May
2013, RT occupied 175th place out of 278 channels in Great Britain, or
5th place out of 8 cable news channels in the UK.”
However, RT also has a social media presence, where it is significantly more successful:
In the U.K., RT’s monthly Facebook engagements (reactions, comments,
and shares) are still well behind the BBC (which gets roughly 14× RT’s
number), but it’s not far behind The Guardian (roughly 1.5× RT)...
In France and Germany, though, RT is a legitimate national player on
Facebook. It’s well ahead of Les Echoes, and Le Monde’s lead is only
about 30%. And in Germany — hold onto deine Mütze — RT was the No. 1
news source in terms of engagements on Facebook in both December and
January, according to this CrowdTangle data.
This is just one platform. But it is also, from a historical perspective, a simply remarkable amount of penetration from a hostile, foreign government. And this is before getting into the 2016 influence campaign.
As for whether this is asymmetrical and to this disfavor of Western governments, my suspicion is yes, because those are the countries that are least likely to simply censor dissenting opinions. China or Iran don't exactly sweat whether or not dissenting views are organic or foreign-backed; they're going to shut it down regardless!