Thanks Cecile!
I did try to manually create a lead without an email address and, obviously, it didn't work. HOWEVER, I have been testing and set up a .csv file with First Name, Last Name, Email Address and associated DSR information leaving the Email Address field blank.
Then I set up a trigger campaign based on the lead being added to a specific list which then sent an alert to the DSR Email Address using a token. This worked, but if the DSR had more than one lead, he/she only got the first alert for one lead.
I thought this was because the email addresses were blank, but all three of my test leads had blank email addresses, but were assigned to two different test DSRs. Each of my test DSRs received an alert, but the test DSR that had two leads only received one.
I'm going to try uploading the test leads assigned to that one test DSR at different times to see if that makes a difference - since my trigger campaign is set to run every time, but that's definitely not the optimal solution.