@King:
If I had a memory of giving a gift to somebody and that somebody suddenly was erased from existence, I would still remember that I gave a gift, I just wouldn't know to whom. We saw that back in the Colosseum with spectators remembering watching Kyros' battles despite not remembering who Kyros was.
It's unfortunate how many people were misled by this, but the reason spectators recognized Riku's fighting style is because it was Riku, their king, who they had presumed was dead or gone. Kyros, not being a blood relative of Riku, could very well have a completely different fighting style, as could Rebecca. In fact, logically, they must have different styles given Kyros taught Rebecca and nobody made any connection between "Ricky" and Rebecca. It doesn't change your point, but I wanted to make the correction.
@Razh:
Brennen assumes a lot. As I said, we really didn't receive any explanation as to how much of the memory is actually lost.
@brennen.exe:
The only frame of reference that comes to mind is Hack/Koala and Boo/Sai/Chinjao. Koala's is the only situation that leaves the possibility of forgotten knowledge given she is at the coliseum, having forgotten communications with Hack. This suggests her priorities have shifted because she forgot about the underground dealings rather than simply forgetting how she learned of them, but it hasn't been clarified. I would think your memory glosses over forgotten persons and attempts to make sense of what happened in the past, and if it can't then you are left mildly confused at best.
Honestly I'm trying to assume as little as possible, given I've used the examples we have as the foundation of my argument. It's true we don't know the full extent of memory loss and how it affects you, but we know people only forget the person, no events distantly related to that person.
The mom in 'The Forgotten of Dressrosa' still had a child and is a parent, she simply doesn't think it's strange. She didn't suddenly forget she had a child because the father of said child became a toy. Likewise the crew still lost to Aokiji and fought the CP9. They didn't lose all of that experience and training just because the cause of the fights are hazy.