Drifting is something that had been defined as 'the place where two Jaeger Pilots share memories and feelings' as 'the place where minds meet.' Everyone had heard the definition from the new trainees to the news stations, but nothing, absolutely nothing anyone ever said would truly prepare anyone for sharing mind-space. Nothing that could be put into words could describe living though memories that belong to someone else as if they are inherently their own.
Hibari and Tsuna go through training together after it is confirmed the two of them are Drift Compatible. They are run through simulations that initiate a Drift between them without the danger of a Jaeger while they adjust to each other.
The first Drift is an experience.
Their supervisor had mention it may be disorienting, that they needed to let whatever memories they begin to share to flow. That the boys needed to 'float'. 'Do not chase the rabbit' was a phrase often repeated to Pilots and trainees- in other words don't latch on to any one memory.
The first time Tsunayoshi touches Hibari's mind he is overwhelmed by a fierce sense of possession. It doesn't surprise Tsuna, when the reason for this possessiveness reaches and flows into his own mind. Hibari had always been possessive of his things and the Kaiju are threatening, not just his town and his people, but his world.
The first time Hibari touches Tsuna's mind there is a huge well of determination waiting for him, He finds he is very surprised. He remembers Tsuna from before- a weakling who seemed to let everyone walk all over him. It seemed that while he had been away the herbivore had started growing fangs, because Hibari can tell that Tsuna means it, that Tsuna will fight with everything that he is to stop the Kaiju. (And an herbivore may fight when it's threatened, when its herd is threatened but only Omnivores and Carnivores will strike first with the intent to kill- and that's what Tsuna will do for his mother, who is threatened by these monsters.)