It's currently impossible to write on signs on Minecraft Xbox One. I can't tell if the issue is with the console or with Minecraft.

The issue seems to be that when the Xbox keyboard pops up after putting a sign down it gets rid of the sign interface meaning you can't write signs on any other line besides the top one. (Pressing START or B backs you out of the sign and the keypad as shown in the video.

^{What should happen is the sign shouldn't disappear when the writing has been confirmed.}