I do not bag my own groceries. The market that I grocery shop has perfectly trained baggers. They even open the egg carton to make sure none are broken and wrap a rubber band around it, so it doesn't accidently open. They put all meats in separate plastic bags and wrap them before they place it in the bag. If they noticed some damage to the items, they'll let you know and then run to get another one to replace it.
Note: if you checkout in the fast lane, 10 items or less, the cashier bags your groceries. The other full checkout lanes have the baggers.