I don't see any area in my profile that would indicate this, unless I'm just missing it
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No, only moderators can see comment flags in the flag queue.
This does exist as an unpopular feature request here.
Generally speaking, the author of a post is never notified of any flag for that post; this means the author doesn't get a notification through the Super Drop-down Collider, or a bar notification, which is the type of notification used to report a new assigned badge.
10k users see the list of flags pending for any post, but:
Just to be sure I am using the right name, this is what I call Super Drop-down Collider.