As long as you realise that implementing this would mean one line in chat every time you hit or get hit (or it gets rather complicated to code correctly). That's a lot of lines, so a good chance to miss messages from your team mates.
Not in favor: again too much spam in an invasion (those are local chat messages, so everyone around you sees them).
Even if this is limited to instances, not sure it's a good idea for the same reason.
And why would you need this? You don't trust the people you go to instances with?
I realize that it would be very cluttered if it was placed in default chat for everyone to see, what I'd like is to see personal damage dealt and received on a mob IF you so choose to toggle this option on. I agree that spam is a huge issue in instances and the like.
I like being able to see what I looted, not what other people looted. I do trust the people I instance with, I just want to have a log for myself for when i kill a mob in an invance/instance/invasion see what I got, not what other people got. We have "item is degraded" or "item is destroyed" messages, why not loot?