America propagandizes China, China propagandizes America, China propagandizes China and America propagandizes America. Or, if you perfer, each country performs "Influence Operations", "Strategic Communications", "Psychological Operations".
Sure. I do prefer the latter terms, though—because using "propaganda" in both contexts makes it seem like it would be the same information being sewn domestically and abroad. Whereas what China wants America to think (and what America wants China to think) are very different things—and pursued in very different ways by very different parts of government—than what each country wants their own citizens to think.
I don't see how using propaganda would imply that the same information are being sewn multiple places.
Propaganda is merely the focused government effort to understand and engage key audiences to create, strengthen, or preserve conditions favorable for the advancement of government interests, policies, and objectives through the use of coordinated programs, plans, themes, messages, and products.
Different audiences are going to require different messaging.
The US's propaganda is quite different than China's propaganda when it comes to the exact physical end product, although the countries are starting to share more and more common tactical forms and procedures.
Of course what the US wants US civilians to think is different than was the US wants China to think is different from what US wants Chinese civilians to think is different from what China wants USG to think is different from what China was US civilians to think.
Propaganda is not a blunt instrument. It has to be and is targeted to specific audiences.
I guess I've always just heard "X Government Propaganda" as an idiomatic phrase used specifically to refer to things Government X tries to get its own citizens to believe on a long-term generational basis, turning it into "common wisdom." Whereas usually foreign-targeted propaganda is designed as part of a discrete operation to achieve some ephemeral goal, and it's the operation that has lasting relevance, not the propaganda itself (which is pretty quickly dispelled as soon as the next generation of foreigners grows up not having been touched by it.)
Or, to put it another way, domestic propaganda is strategic and continuous (and therefore easily observed, by meeting a citizen of the country who "grew up with the propaganda"); foreign-targeted propaganda is tactical and one-off, and won't usually leak out of whatever domain it was focused on (government, military, investors, entrepreneurs, etc.)
It is incorrect that foreign propaganda is never sustained.
For example the United States government is currently engaged in a blanket of propaganda covering communications, social media and news outfit over the Middle East - they go so far as to create video games for the children there to play. The goal is a deradicalization of the Middle East - the vast majority of which now have turned against or are deeply suspicious of the United States (even NATO member Turkey).
How episodic propaganda operations are depends on the goal of the propaganda. Sometimes there is a specific ephemeral purpose (e.g. military deception), other times it is a short term goal (e.g. increase trust in gov't bonds) and other times it intends to be permanent or long lasting (e.g. America and Western culture is friendly to you and you would like to be a part of it).
http://www.au.af.mil/au/awc/awcgate/china/09_04_30_infl_ops....
America propagandizes China, China propagandizes America, China propagandizes China and America propagandizes America. Or, if you perfer, each country performs "Influence Operations", "Strategic Communications", "Psychological Operations".