Well Jessica,
There are two factors that determines this fact as far as I know. Factor number one is depending on the level of attainment of the original emanator and if his attainment is very high, the level of his emanations will instinctively know they are different and will not need guidance and no matter how difficult the situation, they will never stray and will always practice the Dharma. Factor number two is the role and purpose of the emanation, if the role of the emanation is to benefit others on a lowly level, they may not need to know they are emanations. Prostitutes, thieves etc don't need to know they are emanations.
Anyway, even if they knew they will not talk about it. Perhaps Emperor Kangxi knew his previous lives, perhaps he didn't. It is not important. I believe he emanated to pave the way for Lama Tsongkhapa's teachings to enter China. Why China? Because in the 21st Century, China would become a world superpower and be of tremendous influence to the world. Since Lama Tsongkhapa's teachings is part of Chinese culture, it will be promoted as part of its culture and will automatically be promoted and bring millions to the Dharma. Check out Dorje Shugden's foresight and planning! It is incredible!