Ok, I found out how the kindle sends logs along with more stuff...
1) Your kindle "identity" is stored in /var/local/java/prefs/certs/client.p12. This file is like the username and password that ties you to Amazon's whipernet. You need a password to unlock the certificate. I found this password and it's not the best password in the world, but I do not know the legalities of releasing it. (With all this PS3 key and suing going on, it's better to be safe).
2) The kindle uses this certificate to authorize you to Whispernet. There it can do stuff like sync books, download books, send logs, etc.
3) (Cool fact) If you find the password and import the certificate into your computer, you can access Whispernet from your computer and send custom book numbers to sync or check if anything new is ready to be synced, etc.
4) In /opt/amazon/ebook/config/ServerConfig.conf, the URLs for various Whispernet servers are listed along with "commands" for doing things. Each entry starts with a "name", "paramaters", and "url" which points to a URL on the bottom of the file. The kindle reads it as:
https://url/name/parameter=value¶meter2=value2&etc
5) To stop the Kindle from sending logs, the easiest (don't know about best) way is to edit the "name" as it won't potentially "hurt" anything else. I wrote a script that does this. It requires a jailbreak, and it does modify amazon files so you need to uninstall this before updating.
Again, someone should log the kindle overnight and see if my hack works.