It's funny - I'm often in charge of booking the stay for large family get togethers and trips, and I exclusively used AirBnb. However, after using their service for 6+ years, they ended up canceling a reservation I had had for months THE NIGHT BEFORE OUR TRIP. I didn't even realize they had canceled on me til our plane had landed.
Turns out, they suddenly had a problem with a misdemeanor I had been charged with 8 years prior when I was 20 (minor possession of marijuana). They disabled my account and said I couldn't rent through them anymore because of my criminal history. I reached out to them and offered an explanation, as well as reminding them that this was an old conviction of a minor drug possession. I don't have a criminal record beyond that, and had been an avid customer of theirs for many years with raving reviews. They still denied me, and I'm still banned to this day. So yeah, they can go fuck themselves.
It was apparently in the ToS that they hold the right to run background checks on hosts and renters alike, when they choose to. Granted, it's not an in-depth check, but any criminal records that are accessible to the public are accessible to them.