Just to clarify, do you mean thin so-called single use bags, or heavy bags that are intended to be reusable? I've never thought about buying and bringing "single use" bags to the store (I'm sure I'd get strange looks if I did), but I would reuse them for trash like I do with the bags they sell now.
It would probably result in less plastic used, compared to the heavy bags they currently sell.
It would probably result in less plastic used, compared to the heavy bags they currently sell.