I run Space Opera, multiple dueling timelines, Gods, etc. so YMMV....
I also run on the borders of, or just outside of Imperial space pretty regularly, so that changes the vibe a fair amount.
The easy version is to just include it in chargen. I have a Freemason equivalent in my setting, and there's a secret "inner order" that does psionic training (all in service to the Imperium and it's illegal but tolerated if you're "the right kind of person" ala a noble in Imperial service) so one character is telepath who was/is a Navy Commander. It's not their primary skill, but has helped pull them out of a jam - honestly the Freemason thing has probably been more helpful, lol. There's another character that isn't an Imperial, and in their culture psionics are embraced as a gift - so while they get some side-eye from various Imperial Authorities, but as long as they're not out there actively mucking with people they'll generally get ignored (especially since they are travelling with "the right kind of person")
Another character, whose backstory was that they a failed bioengineering attempt by a cult to create a Messiah (cue some stuff RE The Orchard from the old Starhunter TV show). The player just thought it was kind of a fun story and had no intention for the character to have psionics or anything like that (he's basically playing a computer hacker). I, in turn, thought this was a great idea, and it has been subsequently revealed he actually is a psychic (an oracle actually), but blacks out when he has visions, and his self-programmed AI companion (his Daimones) has interpreted his instructions to "take care of him" rather liberally and has been actively keeping this information secret from him because the character "is not as sane as he thinks he is" - incidentally he's also now found out that he is actually a clone of the original character because he's died at least once and started play one time with his Daimones doing a fidelity check...
Still a third character is a field researcher for the Ancient's Foundation, from a minor human race with strong psychic powers and was directly recruited into working for that institution as the Ancients and other Precursors were often known to utilize psionics. Outside of Intelligence circles, the Ancient's Foundation (or related Megacorp versions of it) is probably the primary repository for psionics in direct Imperial service. After that many of the big noble houses have at least some interest in having a psychic or two on staff as well.
My campaign style is that psionics are a tolerated and even an actively embraced skill/talent with certain echelons of the Imperium, but is treated much the same as a technology of mass destruction. Study (let alone use) is highly regulated, carefully controlled, and greatly feared. I also have things like disposable telepathic shields for the military (think of like the equivalent of a gas mask) and a lot more truly terrifying psionic threats outside of the Imperium that drives both the fear of psionics but the need to harness it.
D.