Laser power necessary to cut/etch paper
I have a 405 or 445nm laser diode that I run at ~1.5A.  

It can easily burn darker paper but has issues if the paper is very glossy or white.  

I did get it to burn white paper once but since it is more dangerous (a lot of the laser reflects so I had to wear special glasses) and it was more of a novelty I only did it once.

EDIT

I know you didn't ask for a schematic.  But for completeness here is the setup I used.  The laser still works and it has been 3 years.