The only thing Compoundeyes actually does is increase the rate at which held items appear. Rather than I explain it, let me link you.
So, to put it simply, Pokémon still appear randomly. Their appearance rates are identical. However, Compoundeyes messes with the item held rates. In the case of Chansey, Lucky Punch goes from 50% to 60% and Lucky Egg goes from 5% to 20%, making Chansey 80% likely to be holding something. The reason Butterfree is my Compound Eyes user is simple: it knows Thief and can steal immediately. Even if it faints the same turn, if it goes first, it can take the item.
Right now, I'm giving it a break and concentrating on training everyone in one trip to a Safari...and I just had a beautiful moment. SHINY TEDDIURSA APPEARED...right after I taught one of my Pickup Diggersby how to use Super Fang. Two bites and it went into the ball without any struggle.
I may not have many shinies, but it's among my few now! And it's my first FRIEND SAFARI Shiny! I have to check later if I got the trifecta: the hidden ability; this Safari has the Healer Chansey and Klutz Audino!