Hibari and Tsuna are slowly gathering more and more of each other's memories and as each new memory is lived through and added to each other's own experiences as if they had felt it, as if it was always their memory as if they had lived though it the effects they have on each other grow.
It starts with the fact that they become protective of each other. Viciously so. It's a common side effect for Pilots due to sharing each other's mindscape- lives, memories, feelings, thoughts with each other. Pilots tend to become something like an extension of the other. They become almost psychically aware of the other- they know where the other is without looking, know what the other will do as they choose to do it, and unconsciously synch their movements- almost as if they are one mind in two bodies.
The second effect comes from the fact they live through the other's memories as if they are their own. It calms Hibari down, but it causes Tsuna's temper to move closer to the surface.
It's an obvious conclusion.
When someone like Tsuna shares a mind, memories and feelings with someone like Hibari, who fought and bled for his place as the biggest badass in Namimori it's hard not to have things bleed over. The same holds true for Hibari who had 'lived' through Tsuna's life and experienced his calm steady temper (which surprises Hibari. It's not that Tsuna is never angry, it's that Tsuna is very controlled) and his nearly endless compassion which bleeds over in such a way that it soothes and calms him (which is new because Hibari had never actually been completely calm before but Tsuna had managed to help him achieve it).
Neither of them had expected it when Tsuna walked up to Hibari, but they find that the both of them were glad they were Drift Compatible.