Is there a way that I can set it to make certain usergroups be allowed to use private messaging between specific usergroups, instead of every person on the forums?
You'll need a plugin/modification for that, I believe.
Do you know if there are any that exist?
No need for the plugin;
Admin CP > Users and Groups > Groups > Edit the group you don't to send PMs > and under "Miscellaneous" tab > edit "Private Messaging" options.