This depends on a couple things.
If a moderator on the site that a room is hosted on is in the room at the time that the message was flagged, they can see who flagged the message. If they weren't there, they can't see it anymore. They can't see it after the fact, and mods who aren't mods on the site that the room is hosted in can't see who flagged it.
So, for instance, if you flag something in The Reading Room (the main chatroom for Literature.SE, where I'm a mod) while I'm actively in the room, and I manage to catch the blue flag before it's handled I can see who flagged it. But if Rand al'Thor is in the room (who's a moderator on Scifi.SE) at the time, and catches the flag, they won't see who flagged it.
But if it's in The Restaurant at the End of the Universe (the main Scifi.SE chatroom), it's the opposite - Rand al'Thor will be able to see who flagged the message, but I won't.
If you moderate multiple sites, and your chat parent is set to one of those sites, and someone flags a message in a room parented to a site that you moderate that is not the site that your chat parent is set to, you can see who flagged it. ...That is to say, Catija moderates both Interpersonal Skills Stack Exchange and Arts & Crafts Stack Exchange. If something in The Studio (the main chatroom for Crafts.SE) gets flagged, she can still see who flagged it, even though her parent site is set to Interpersonal.SE. She can also see flags in The Awkward Silence (the main Interpersonal.SE chatroom).
And of course, CMs have ways of finding out who flagged a message, even after the fact. (The majority of users won't be able to see that link. It's a mod-only link, sorry. But it's a response from a CM that clearly indicates that they have a way of seeing who flagged a message when necessary.)
Regular mods can see a list of which messages have been recently flagged, and the number of flags it received (including flags (includes 'valid' clicks), counterflags ('invalid' clicks), meh flags ('not sure'), and custom mod flags). That list is located at
/admin/recent-flags (link 404s for non mods). This list doesn't include the information about who flagged it, though.
Custom mod flags look a bit different. They show up as orange or red (I'm still not sure when it shows as orange or red, it seems inconsistent), and these always show the name of the flagger: