Moderators have access to a page which shows recent chat flags from all the rooms on a particular chat server. This is a useful tool, however we cannot see who dismissed any particular flag. With a moderator flag, we only know that a mod would have to be the one that dismissed it, but have no other information than that unless that mod goes out of their way to give that information to you.
This isn't idle curiosity, chat moderation in particular is often a team effort that involves talking to moderators from other sites. Knowing who handled the flag would help this conversation from both sides.
For example, I recently let the moderators of another site know that a flag occurred in their room. I did this because I wasn't sure who dismissed it and it contained information that they would find useful. If I could have seen who handled the flag and saw that it was a moderator from the site I informed manually, I wouldn't have bothered.
On the flip side, if a flag occurred in a chatroom on a site I moderate and I see a flag was dismissed by another moderator and that confused me, seeing who did it would allow me, as the local mod, to be able to ask for more information from them and determine if there is anything else I need to do.
So I feel that this information would be a helpful addition to the information we already see on that page.