If I were to tell you a "fact", after which you would go online and check it for yourself, only to find the information I provided to be untrue, would you ever believe me again? No.
So, by that reasoning, evil propaganda shouldn't work.
By the way, propaganda by itself isn't a bad thing. There is "propaganda of common sense", and "propaganda of healthy lifestyle" - being honest about your attempts to change people's minds is not a bad thing.
The way bad propaganda works is not simply by lying to you, the point is to make you feel like you can never say what you think. So by that logic, Kanye West will be alright in the end, as he is openly saying all the crazy stuff he believes in.
The best way to "teach" people propaganda is by example. The propaganda show will not explicitly say what it wants to say but will keep dancing around the bush with all possible tangents until the shape is clearly defined.
That is why people who are exposed to it for the long term tend to see a hidden subtext in every sentence they hear. Sure, even straightforward public statements have more to them if you dig deep, but the propaganda wouldn't go into analysis, but instead imagine every word to be a Freudian slip and every joke to be a subtle taunting or a nod. Especially if the statement is direct, curated and not fueled by clear emotions.
This is evident by the fact that propagandists, despite often hunting for those Freudian slips, are more likely to look for them in prepared statements rather than in live improvised talk shows, where it is easier to make mistakes.
This will also make the person watching fearful of expressing oneself without emotional context. This in turn is evident by the fact that affected people often express themselves in the context of anger.
Another component is "power". You will not find propagandists using Warhammer 40K Lore to push their propaganda. They always "attach" themselves to things you can't detach from, like politics & economy.
If you feel like there are thoughts that you can never express to society out of fear of it not accepting you, then you are probably affected by evil propaganda. Manipulative bastard's main goal is to seed fear into a person, fear of society, fear of working things out.
Victims of such manipulations (a.k.a. russians, MAGA) are often extremely patient, saying just the right things, even too right at times: "I'm against all violence", and "Life is the most important thing". Good things to say in general. But when you are attacked on the street, you'd want friends, ready for violence, by your side.
Be brave, and say what you think, always. In the end: nobody really cares what is on your mind, so why give yourself a headache to hide it?
How to save people from brainwashing
My experience is based on a family member of mine:
Lead by example. Make sure that everything you say is more "out there" than whatever the other person might say. Hearing some xenophobia? Feel free to talk about aliens and X-Files) Or whatever feels more right to you.
Instead of trying to be fearful of saying the wrong thing, be bored of empty statements, and always get to the point. Boredom is the best tool in general. As hooks of the brainwashing are through "being very important" and not by actually being entertaining.
But don't assume yourself to be grounded and the person to be detached from reality. Make sure to be the one who chews freely through every idea and topic in the conversation. Make assumptions, admit your mistakes, and then make a new one. Talk about things in depth until they are exhausted and boring, and never run away. Talking about things is never harmful. Just be honest - it is quite easy if you get used to it, it doesn't require much mental effort and keeps your mind free to do other things.