How do we get our kids to sleep independently?

Kids are 5 and 3. The youngest one needs my wife to sit beside him and pet his hair / back to fall asleep. He wakes up at least 2 or 3 times a night SCREAMING. My wife has to wake up and go comfort him back to bed.
Our oldest one was never sleep trained, he NEEDS to sleep beside my wife. If not, he screams and cries. If he wakes up and she's not there - the screaming and crying begins.

We put their beds in 1 room last night and bought them one of the "sleeper" clocks that turns a color indicating that it's either bed time or "awake" time. Our goal was to lay them both down and for us to sit in a chair in the middle of the room while they fall asleep. Our oldest one actually laid down relatively easy (didn't fall asleep), but the youngest one threw the biggest fit of his life. Screaming and crying, hyperventilating, he gagged and almost threw up at one point.

We wanted to try the 'silent' treatment where we tell them that it's bed time, and if they cry or get out of bed we just put them back into the bed, but this prompted the biggest freakout of our youngest one's life. We thought we'd tough it through, but it went on for about an hour. AN. HOUR. They don't just cry or call for us, they legitimately wake up screaming and crying. This, of course, makes us feel like terrible parents and we ultimately capitulate.

Part of me feels like they know that they get what they want if they keep throwing the fit. But how long do we let them scream and cry (literally to the point of almost throwing up) to where it just borders on neglect? Like, we're sitting right there in the room with them. They have waters. We do bathroom and bed time routine before bed, etc..etc..

We could probably endure if it was just a little bit of crying and whimpering, but the moment they lay down without one of us and we try to walk away, It's the biggest meltdown of their lives. Help.