I'm iffy with traded Pokémon. On one hand, I found a shiny Bidoof in my boyfriends game and he let me keep it, and even though it has his OT and ID, it still feel extremely special to me. On the other hand, I don't like using traded Pokémon, even if it's one I caught/bred myself, if it doesn't have the OT as the game I'm playing then I won't use it. No real reason, especially since they rarely level up enough to stop obeying me, I just enjoy using my own Pokémon.
As for GTS, I've traded on GTS for Foreign Dittos/Pokémon for breeding, or to help fill up my national dex, but I'll never use them again, because they're not my Pokémon. I even got a legit Japanese shiny Butterfree, the only shiny I ever got on the GTS, and I just keep it to look at.
TL;DR I only really use my own Pokémon but sometimes Pokémon traded from friends can be special! GTS Pokémon don't mean much to me though.