Why would public repos on github effect my private repo on bitbucket?

Exactly. Roll your private repos on bitbucket and public repos on GitHub for the exposure.

What's the difference between publicity of repos on Bitbucket and Github?

To me, if the task is to see the code, they seem essentially the same — a storages one can checkout from, with a nice-looking web interface to preview the contents.

As for collaboration, DVCS are distributed by their very definition. Sure, clicking on "fork" button on web is easier that "pull; remote add; push" series of commands, but I believe this is completely negligible threshold as compared to actual work.

GitHub is more likely to pass the "dumb HR" test. "Oh a GitHub profile. I have heard of that."

I think one can just write "Git" instead of "GitHub" and point anywhere.