The room a flagged message appeared in is urgently needed for deciding whether a flag is valid or not. Java bashing would be offensive in the Java room, but might be fine in the C# room, so when I see a message bashing Java, knowing the room is absolutely necessary to know whether this message is offensive.
However, as Anna pointed out in her comment to the question, the room alone is not enough, usually you also need the textual context to decide and often you'd need to know room memes and policies, too.
But when I am in the chat, I am there for recreation. (It is called Lounge for a reason, when I want to discuss facts I can go to SO proper.) I am not eager at all to follow half a dozen links into rooms I don't know, trying to understand discussions about topics I don't know about led by users I never ran into in my little ecosystem on SO.
So in 90% of all cases, I click "I dunno", just to get that annoying reminder off the lower left corner, and the whole flaggingsystem just doesn't work for me. From what I know from other users of the C++ room that they feel the same. The notifications are just annoying users.
So here's what I propose, in the order in which I think it should be implemented:
Add the name of the room where the message was banned.
That's fairly easy to implement and would somewhat help immediately.
Let messages "expire", so that they cannot be flagged after a while anymore. That would prevent stupid cases where some kid goes on a rampage and flags old messages. If that leads to a really offensive message to slip through the net, one can always flag for a moderator.
I suppose this is easy to implement, too. ICBWT.
Allow those asked to verify a flag to see the flagged message's context. Ideally, I could chose between seeing the five messages before plus the five after the one flagged, or seeing the one it replies to (if applicable) and the ones replying to it.
That's probably harder to do.
Allow/require users to give a reason for flagging. Is this message offensive because it bashes Java in the Java room or because it's a picture that's offensive to others? Is it spam? Is it someone drive-by linking their question (dropping a link without participating in the discussions)?
I have no idea how hard that would be to implement.