How do Military police officers compare to local Police officers?

Not much. Both receive training in firing weapons and self-defense and police tactics. Military Police are trained to guard military facilities and prisoners of war, whereas civilian police protect your home and private property and arrest and guard criminals.
--- I actually disagree with the above answer. It does depend on what branch of service you are asking about as each branch does things differently. However, as a former Air Force Security Forces member ( the Af's version of MP's), we are actually a lot like civilian police officers, if not a little more advanced. True we both attend basic training in weapons and police tactics as well as self defense, much of which is similar training. However, (speaking specifically of SF's) we are also trained in base security measures, ground combat measures, convoy maneuvers, heavy weapons training, hostile situation training ( i.e. base attacks). In addition to this training we are also trained in the local Law Enforcement procedures - arrest/detention; searches of people, buildings, vehicles, and property; questioning tactics; defensive/offensive driving..ect. So, at least as far as SF goes, we can do anything, if not more than, a civilian officer can do in terms of training and abilities. What we do not have is the authority to exorcise such training off the installation unless we are dealing with another military member. Most often, for a civilian officer to accomplish the training that we receive they have to attend specialized training in the field of interest, e.g - SWAT members usually have to apply and try out for the SWAT team and go through specialised training, where as..SF/MP's are already prepared in training to handle such situations. However, on most installations there are special teams formed for doing SWAT maneuvers so that other team members may focus more on the job(s) at hand during a crisis situation. The only differences between SF/MP's and civilian police officers is the difference of pay ( often civilian police officers start out at a much higher pay rate than any starting enlisted military member), and the risk they take on by wearing the uniform. Civilan police officers face risks every day in many ways, and sadly some do not make it home at the end of shift. SF/MP members often face some of the same risks as well, just not as frequent. They both face bank robbers, drug dealers, armed domestic situations, felony traffic stops, and just people simply not paying attention. However, SF/MP members are presented with a much larger threat that the civilian police force understands but rarely realizes how close that threat often is. SF/MP members also deploy into a much larger threat, where as civilian police do not. Ultimatly, in the view of a prior service member - the police protect the civilians from harm, while the SF/MP's protect the police from harm.