When I'm inviting people to share a document with me, sometimes I put information in the invitation body that I want to share with people not on this invitation list. For example, I may have already shared it with them, or I may want a copy of the email for my own records.

So it would be nice to be able to add email addresses to whom a modified invitation email will be sent: in place of the #i link it would say something like "Andrew has invited N people to share the document Such-and-such (with a #d link if the recipient is already on the sharing list) and included the following text in the invitation email:"