(I hope they also let you do it yourself if you want to avoid stuff like this?)
Some stores do have a self-checkout (e.g., Meijer), but it takes about twice as long to check out that way, because you are not the cashier and don't spend forty hours a week running items across a scanner and tucking them into bags and consequently haven't attained the same level of deftness. Also, paying takes several times as long in the self-checkout, because you're using a slow, badly-designed touch-screen interface; whereas, a cashier in a regular checkout lane would be using a traditional keyboard-based register.
Every once in a great while (about one in a thousand times or so, I'd guess), your regular checkout cashier will be a fresh green trainee, and it'll take about as long to check out as if you used the self-checkout. When this happens, it is REALLY annoying.
When I was a kid, the cashiers ran the items across the UPC scanner (or punched in the price manually -- not every store had the barcode readers yet back then) and then set each item on the far end of the checkout counter, beyond the scanner. A separate person -- usually a store employee -- would then place the items in bags. This required two people, however, and it wasn't faster. If anything, it was slower, as bagging tended to be an extremely high-turnover position, so the likelihood of getting somebody who was new at it was higher. Every store I've been to in the last fifteen years or so has switched to the system where the area just beyond the scanner is a bag rack, and the cashier runs each item across the scanner and places it in the bag. This takes roughly the same amount of time as setting it on a counter, and you don't need a second person to put the things in bags.
So, if you want to bag your own stuff, you'll need to use the self-checkout lane -- which, as I said, takes about twice as long.