Has anyone ever figured out for sure what it is the hero and heel badge do? I mean, I know it makes the monster bad or good. But is there definitions on how it does it?

Watching my SO play his game, I have one suggestion: they affect the monster's actions in training. While he presently has a Metalner that is only 'bad', it's cheating easily as much as any 'Worst' critter, and possibly even more. Therefore: is the fact that he has two heel badges affecting that? Giving more chances to approve or disapprove? Could the herpo badge do the same with 'Great'?

He also suggests that the badge may effect the monster's reaction to being scolded/not scolded. Heel badge makes them less likely to react by turning good, while hero badge does the opposite.

Any thoughts on this?

Yep, that's pretty much what it's been like in my experience, and I've had a LOT of experience warping monsters' personalities for better or worse, considering my obsession with getting all techniques for each monster type.

A little bit of research by Nevistar in this topic indicates that the Hero/Heel Badges DOUBLE any positive/negative changes in nature, respectively.

Example: If you praise your monster for doing good work, the monster's nature will increase by 33. If you have the Hero Badge however, the increase will double to 66.

Likewise, if you let a monster get away with cheating, the nature will go down by 33, but if you have the Heel Badge, the decrease will be 66 instead of 33.

That's the known effect of the badge's so far.
Speculation: Having BOTH Badges will not neccessarily cancel each other out, but amplify the effects of praising/scolding (of the lack thereof) on the monster nature-wise.

Again, it's nice to know the raw statistical data after playing by (sometimes incorrect) instinct for so long. Thanks again, guys!