I catch them on my island or island tours, but never on a foreign island. Once, I caught two new fish on a friend's island who has literally the same time zone as me, BUT I still went back to my island and I grinded for 3 hours trying to find the same fish locally so I could donate them to my museum. Yup. I'm that crazy...
I catch all of them on my island or my S/O's. Scaring off other species of bugs/fish is a lot more efficient with another person. They can also help you look out for whatever you're missing.