You should wash your hands after changing a diaper.
When changing a diaper you can get germs on your hands and then spread the germs around.
So always wash your hands after changing a diaper and especially after changing a poop diaper.
I admit there have been times when I have not bothered or forgot to wash my hands after changing a diaper and I've been okay.
However I wash my hands regularly anyway and especially before cooking food or handling food.
But you should ideally wash your hands after changing a diaper.
In a daycare setting you have to wear disposable gloves and also wash your hands and the kids or babies hands when changing diapers.
But at home you don't have to wear gloves unless you wan too.
I do however wear gloves when I change my kids poopy diapers if they are really bad such as having a poop explosion or diarrhea in the diaper.
It's not because of germs or I'm afraid of getting poop on my hands but instead it just makes cleanup easier and it's better to avoid getting the poop on my hands to have to deal with cleaning off later.
After changing a diaper the babies or kids hands should be washed or at least wiped down with a baby wipe as well.