Hi Sue,
I've read the MZ privacy statement and it says: "Each badge you push to the Mozilla Badge Backpack contains your Persona username and data about what the badge means and who issued it." It does not say your Persona email address will be published in order to ensure you are who you say you are.
I don't think that their privacy statement is very clear about this, but a "persona username" is an email address (see here - it says "usually" but I'm not sure how it could be anything else). You can also see in their terms of use:
Badges may only contain the user’s email address, the issuer’s name, the badge name and specific data about what the badge means such as the badge description and criteria URL, which is a URL pointing to a page hosted by the issuer that details all the criteria required for earning the badge.
However despite this, I agree that at the very least we should give the user a choice about if their email is displayed, and ideally avoid displaying it all together.
The only reason I'm giving any push back at all is that we also want badges to be verifiable. To be clear, given a choice between good privacy and verifiable badges I would still choose good privacy, but what I'm wondering is, is there a way for us to have our cake and eat it too. 
Unfortunately as of right now I can't think of another way, and ultimately this is a problem that needs to be solved by the badges infrastructure rather than us. If anyone has any thoughts then please share!
that we have a unique email address for all learning experiences and that it will stay with us forever and ever
Again I agree this is definitely not the case. Unfortunately this is a limitation of the current badges infrastructure. I have already been involved in the discussions with Mozilla about this issue here:
https://github.com/mozilla/openbadges/issues/42
Unfortunately there hasn't been any progress on this issue, and we've done just about all we can to mitigate it in moodle by allowing users to at least pick which email they connect with.
Anyway, I will discuss some more with Yuliya and we will try to do something soon.
Simon