I bought several from the grocery store I shop in the most. They look a lot like the flat bottomed ones from Amazon - grocery tote bags. They've got the grocery store logo on them and were half the price (99 cents each) of the Amazon ones. I feel a little squeamish taking them into other stores, but I do. I've taken them into Sam's, Target & another grocery chain. One of the managers of the other grocery chain was bagging my groceries and offered in a friendly kidding way to cross off the competitor's logo and hand write his store's name in its place. I like the idea of having a collection from multiple stores. I may do that as I need to buy additional or replace bags.
I really like using these. You can fit a lot in them and don't have to make a millions trips to bring in groceries in the plastic sacks. And of course I get to feel better about not dumping additional plastic bags into the environment. The rigid plastic bottoms mean you can carry a gallon of milk or a large juice container or a bunch of cans in one. You can make them so heavy you'd prefer not to carry them. I've also used them to tote other bulky non-grocery stuff around. These do have handles that are long enough to go over my shoulder.
I have a large insulated bag (you can lay it down flat and fit two large pizza boxes inside to carry the pizzas home hot - not that I would be involved in eating any of them once they got home.

Or you can stand it upright and pile in the stuff you want to stay cold on the way home from Sam's or the grocery.) I don't take it into the store. When everything is empty, I fold up the grocery bags and store them inside the insulated bag.
(By the way, when I saw the thread title, the thought crossed my mind to hijack the thread and ask how people felt about taking bags into competitor's stores. So I was pleased to see that was part of your post.)