I dunno. There's no sign about it anywhere in the restrooms at my office. They are just regular in-the-wall bins which they line with plastic bags. They take out one plastic bag and put in a new one. It's not like they are handling the trash itself. Plus I'm not just dumping the item in as is. I wrap it up in TP first.
Honestly, I've never heard of an issue with blood products being thrown in a trash can in a public restroom, nor would it have ever occurred to me to think of it. It's a restroom trashcan. I'm sure all kinds of trash makes its way into that bin.
Not to mention that I'm pretty sure that the same person who empties the bins by the sink is also emptying the paper bags in the stalls. I'd be pretty surprised if the paper bags which hold nothing but these items are any safer to handle than the large plastic bags where my trash is buried somewhere in the middle.