Thank you for your insightful additions, Gustavo and DocV. In revisiting this thread, I have decided that I have more to say.
Your example "He would know. He is a senior official in Trump's administration." is clear for me to understand. My problem is when someone says, "I would know." Further explanations would be appreciated.
It works the same way, Fujibei. It has to do with the "posturing" I was talking about. You are capable of thinking about yourself as Fujibei, right? You can adopt the position of someone else looking at you or listening to you or considering your credibility in light of facts about you.
When you say "I would know" in the way in which it is being used in these types of sentences, you are using "I" to refer to yourself, naturally, but from the standpoint of your interlocutor, from whose standpoint you are "he" (assuming I have guessed your gender correctly).
If you can understand "He would know. He is a senior official in Trump's administration," you can also understand "I would know. I am a senior official in Trump's administration." If you used "I would know" like this, you could replace it with "Fujibei would know." You could speak of yourself in the third person.
Interestingly -- now I am getting to the main reason I have made another post in this thread -- we also use "I wouldn't know" in a similar way. Indeed, there are no fewer than 548 results for "I wouldn't know" on COCA. Here, too, one is positioning oneself outside oneself for rhetorical effect:
A: I wonder if prison beds are comfortable.
B: I wouldn't know. I've never been in prison.
A: I wonder how the band sounded last night.
B: I wouldn't know. I wasn't there. But DocV was there. He would know.
In both examples (especially the second), "I don't know" could, of course, be used instead of "I wouldn't know." I say "especially the second" because, in the first example, "I wouldn't know" packs a nice rhetorical punch. It's a bit like saying "Don't ask me." It has a slightly edgy quality.