Private Auras are back in The War Within Alpha, and they now affect Mythic+ abilities as well!
Private auras are a flag that Blizzard is able to set on spells in World of Warcraft that only allows addons to see basic information about the spell. For practicality, it basically prevents assignments weakauras from working automatically.
Private auras have been a controversial topic ever since their implementation in Dragonflight. Some private aura mechanics have been so difficult in the time that the encounter gives you, that you need to use a computer assignment. This has resulted in the Macro system being used, where players hit a macro to indicate they have a mechanic and a Weakaura does the assignments for the players. Many players have expressed their dislike of this system, as they've turned some progression points on bosses into a Weakaura and macro debugging phase rather than actually progressing on an encounter for the exact same result as before.
In the War Within Alpha, it seems that a round of Private Auras have been applied to spells that indicate that a mechanic is targeting a player. This is pretty standard for raid and didn't cause too much of an issue except for Volcanic Heartbeat on Neltharion, but Private Auras are completely new to Mythic+ dungeons. The impact of these will remain to be seen as it's completely determined by the complexity of what the encounter is asking these players to do and how important it will be to drop these mechanics in specific locations. If it's like Rending Charge, these should be not noticeable for most players. However, if it's like the original Volcanic Heartbeat, this will likely be something that many players will give feedback on.
These encounters have not been tested yet, but it's likely that Resha'nan in the raid will be similar to the dungeon version. See the Dawnbreaker section below for more details on the dungeon version of the mechanics.