The only options to share my photos that I have are:
Who is allowed to share your photos or video?
You can set a preference that determines whether or not the "Share This" button will show for other people on your public uploads - it will always be available for you on your content.
The button will never show alongside things that aren't public. (Note that even if you hide this button, people will still be able to grab a URL from the browser address bar and send that around.)
Your Friends and/or Family
Any Flickr member (Recommended)
I have it set on "Any Flickr member (Recommended)". That seems to be as about as public as I can get.