Some think he’s a myth, so I’m sure rumors spread and differ depending on how often and directly criminals interact with the Bat. Petty criminals may wonder all sorts of things, considering he’s so skilled that he may seem like more than a man, whereas supervillains who face him with regularity are more likely to know his abilities and limitations.
Interestingly, the idea that criminals may wonder whether a hero is just human or has superhuman powers is a plot point of Kobalt, a title from Milestone Media (later bought out by DC) starring a hero of the same name who was essentially Milestone’s take on Batman, although Kobalt may have been even grittier (or par for course considering his comic released during the ‘90s, a period known for grit all around).