It depends. If you are playing in a group that has charts for each horn, you just read the music. Some folks do stick with one horn exclusively, but many will play multiple horns. I play alto, tenor, and baritone sax as well as clarinet, bass clarinet and flute (poorly). Most of my work is on tenor, but if I get a call to fill in on another horn, I am there. If you want to do musical theater, you need to play multiple horns to cover the book. In some cases that can expand to playing the double reeds (oboe, English horn and bassoon) and other exotica as well. The last show that I played was Nine to Five. The book I played (the second part) required tenor & baritone sax, clarinet, bass clarinet and flute. The first sax book was alto sax, clarinet, flute and piccolo. The woman who covered the part didn't have a piccolo, so she covered those parts on the flute, playing it up an octave where possible.
If you are playing from lead sheets or fake books in concert key, you quickly learn to transpose on the fly. This is (to me) easier on a Bb horn since you are playing one step above the written notes. On an Eb horn, you need to play a minor third below or a sixth above the written notes. That's a bit more difficult, although many pros do it all the time.