When cutting an offshoot of lucky bamboo, is it necessary to cut the main stem afterwards? What will happen if you don't?

Also how often do you water the offshoot (mine is growing in soil)?

No, there is no reason to cut the main stem when removing the offshoot. The main stem may produce a new offshoot at the same location or at another node.

Offshoots grown in soil should be in very small pots with a porous potting mix that you keep barely damp until the roots system has established itself.