NPC responses during the flagging process get split into two different chat filters. The first is the typical 'say' filter that the majority of responses will be filtered to. The second filter is the 'other' filter. The 'other' filter will have the responses from Seer Mal Nae'Shi as well as all the responses where you get an actual flag. Make sure you can see your window that has 'other' filtered into it to ensure you are actually receiving your flags.
At the end of certain raid encounters A Planar Projection will spawn. We will need to hail this projection for a flag. DO NOT hail the A Planar Projection more than once. Also, DO NOT hail until the raid leader calls for it. Failure to follow these rules will result in being removed from the raid for the night.
Allakhazam Planar Progression Guide
Click the link next to each tier to see the actual flagging quests.
To check your progression flags you can speak to Seer Mal Nae'Shi in the Plane of Knowledge.
She is located at -40, -200 in the building near the Plane of Tranquility stone.
Make sure you are sitting when attempting to talk to her. First say "unlock memories" until the response you get is repeated. Then say "guided meditation". Use the following resource to check what these responses indicate.