Using embedcode from List Media API call

Hi Chris2020. Sadly, there isn't a lot of documentation on this - it's a bit of a legacy feature. That said, what you're doing should work, but the media being embedded would have to otherwise already be viewable by anonymous users according to the permissions applied to it. That is, if you are logged out of your community and you can't view the media's page in its gallery, then it also wouldn't be available to view when embedded in another domain. 

Is there any way that I can leave the file on a telligent server and serve it up at a different domain without the user being logged in at telligent?  Or will I need to set up a proxy?

Any embed will be subject to permissions regardless of storage, for it to work you would have to have some sort of embed or url separate from the community.  But if you are willing to do that, you might as well make the post publicly accessible anyway.