I've seen claims on them that they are broken down by sun light.
Regardless I throw my soft plastic in the recycle bin for soft plastic and before we had one of those I used the "rest garbage" bin instead because back then it wasn't recycled anyway so regardless of which bin you used it would get burned.
Plastic bags hold better and hence they are better for me.
What I normally use as a grocery bag is a 4 SEK IKEA Frakta Medium bag though, often lately I haven't brought it and taken those small fruit bags but that's pretty stupid, but if I don't have anything else and don't want to pay for a larger stronger plastic bag that's what I do.
A regular plastic bag is like 1.5 SEK, the Ikea Frakta Medium is 4 SEK as said. Paper bag is likely at least 2 SEK.
The Frakta holds way better and is about the size of the paper bags.
$0.9 in the US.