In my experience, usually the cashier bags, but yesterday at the supermarket, there was a separate bagger.
With everything else we’re doing to try to minimize spread, I would have thought that bagging your own groceries is not a major hardship.
A lot of people are asking me to bag their own groceries and sure, we’re happy to let them do it, but most people are slower than those making a living at this sort of thing.
Also new hires that are slow, people who aren’t new but aren’t motivated/have a poor attitude, and probably other reasons I can’t think of at the moment, what with just getting off a work shift.
My town did, too, but due to the pandemic they backtracked. Reusable bags are currently illegal, and the supermarkets are no longer charging for cheap disposable bags.
I’d have thought that asking us to bag our own groceries (in our bags or their bags) would be safer, but the markets are still bagging my stuff. Possibly because the professionals move the line along faster than shoppers would.
I glanced at a few of the comments just now. I agree that they’ve got it tough. Especially in the early part of this, people went nuts over TP etc. and I would have hated to be on that front line.
In the situation I described, none of the the people I mentioned were wearing masks, gloves, etc. but I had my mask on. The lines haven’t been terribly long lately, however.
There’s another cashier over there whom I used to actively seek out because she was good. Although I saw her the other day, I think her good work got her promoted so now I have to find someone else who’s dependable. 
But, good for her.
As I said, in the Southern suburb, it happened so often, I had no choice but to bag my own groceries. In the Northern, it was the first time it ever happened.
In the Bay Area they banned reusable bags but also did away with the charge for plastic bags. My Safeway hires mentally challenged people as baggers, and they are still there. I prefer to bag my own stuff but don’t want to do anything to put their jobs at risk.
If not bagger is there, I bag myself to move the line along. The checker bags whatever is left. But the new plastic bags are too small and often a pain to get open. I’m looking forward to using my own again.
Never ran into the OP’s problem.