Whisker chewing can be caused for different reasons ...
The alpha chin could be showing dominance by chewing off the whiskers of the less dominant chin.
Jealousy ... one chin being mean to another for no apparent reason.
If the chinchilla is chewing it's own whiskers ...
It could be nervous, stressed or scared of something {maybe even it's cage mate} ...
It could lack certain vitamins or minerals.
A bad habit that needs to stop. I would think about separating them.
How old are these chins?
How long have they been living together?
Is this a new occurrence or has it happened from the beginning?
Are you sure one is chewing it's own whiskers, or could the other chin be chewing them off?
Do you give them extra calcium?
Is there any chewing of the fur?
Sorry about all the questions, but the answers could help to solve the problem.
Jo Ann