I doubt there are statistics on this, but I was wondering if other information could answer the question. I was reading about how police (at least in the modern understanding) haven’t always been a law enforcement practice, and I know during the time period in question public punishments were the norm, but I’m wondering if we have any way of knowing how effective this method was at preventing crime. If possible, I’d also like to know how crime rates then compare to current crime rates, and if the types of crimes commonly committed (I.e violent vs. non-violent) differed as well.
Well a great deal of the information that you're looking to find is just not available. Things like modern crime statistics just simply did not exist in the Medieval world. We quite simply have no idea what portion of medieval crimes were "violent vs non-violent" (which is itself a modern distinction that the Medieval Mind would not have really considered) nor do we have knowledge about deterrent punishment and its ability to actual deter criminal activity.
Now part of the issue is that even our medieval law codes don't really give us a clear picture of what the situation was like. We know that for example in Anglo-Saxon England there were quite severe laws concerning theft, especially under rulers like Æþelræd, but we don't know if the repeated issuance of law codes that condemned thievery was a response to a huge crime wave that swept the land, a rhetorical tool, or something else.
Now we can say surely that Medieval people believed that public physical punishment served a deterring factor in law enforcement efforts, but we cannot firmly back that up. It is also worth mentioning that the nature of public punishments likely varied significantly based on location, time, and the offense committed.
I've written previously on aspects of crime and punishment in Anglo-Saxon England if that might be of interest to you!