My first shiny encounter was in Silver version on a school morning. I was just sitting in the living room roaming around outside of Mount Silver when I ran into a shiny Ponyta. I had gasped and stared at the GameBoy screen in shock (I might've ran into the kitchen to show my mom, she was directly across the room from me), and then I decided to weaken it to catch it. Then it fled. I was obviously pretty upset.
About ten years later, I'm in my room with my brothers watching something online, and I'm training in Mount Silver. I was walking around outside when I ran into a shiny Rapidash. I poisoned it, and just managed to catch it on its last turn. Then about several weeks later, I'm training in Victory Road on my Diamond version, and I run into a shiny Onix. But then I KO'd it... somehow. I did an overly-dramatic "NO" in the kitchen before I went about my normal business.
So it's safe to say that both times, I've gotten very excited and nervous. Chances are I'll continue to remain excited and nervous each time I do encounter a shiny.