They do get access to some countries' criminal and arrest records. I was arrested 15 years ago for shoplifting in Hamilton (New Zealand) in spite of my friend vouching that I had in fact paid for the goods, but I'd lost the docket.
When I went to court I was able to prove that I had in fact paid and my case was dismissed. Before I t took my family to Disneyland 20 months ago I'd been told I didn't need a visa if I had no convictions and it wasn't until I saw the wording of the form on the plane that I realised I might be in trouble, as I had been arrested but not convicted.
I answered No but on passing through immigration I was put in a room and quizzed about my background for four hours. They constantly asked me about that case, but nothing I told them seemed to make any difference. We were placed on a return plane to NZ, our holiday ruined.