Do companions have seperate chances of getting detected, or is it all up to the player character? And what behavior settings should I keep in mind when sneaking?

All up to the player, you get seen, they get seen

quite funny seeing a companion right next to a hostile character and neither do anything until you (the player) are detected. (passive mode)

I've had my companions get detected before I did. I stay in [CAUTION] mode but (for example) the Fiends in the distance are opening fire on Cass and Ed-E. My companions have cost me my full [HIDDEN] status on numerous occasions.

edit: if you're going to sneak through hostile territory, leave your companions behind, where they can't cause you trouble. As for your own behavior: make use of cover and shadows, and wear light armor.

Whatever is 'technically' meant to happen, I'd still advise NOT taking Boone Or Ed-E when trying to sneak past a family of Deathclaws, or a pack of Cazadors. You can edge and sneak as carefully as you like, you know you're going to hear that blasted robot play it's tinny battle tune and zoom off into a cluster of teeth and claws and death for all. Perhaps it was a bug in my gamesave, perhaps Ed-E is just a damned stubborn machine, but regardless of what I tell it what to do and what not to do, it just has this Deathwish in regards to the predators of the wastes that just love to use it as a football.

Actually, I find that setting ED-E to melee mode prevents it from attacking at all.

a high karma & perception will cause companions to do this more frequently at longer distances (as i found out), but i dont mind, ive got a very high sneak skill (100), so i just enjoy watching all 7 of them (im using the PS3 companion glitch) fight 6 death claws, companions set at melee while the death claws run right past me as if im invisible (crouched) lol