I had a Skype interview with the hiring manager (who became my supervisor) first, then a secondary Skype interview with the publisher. I made sure to email thank you's to each person after the interview (going as far as to ask the hiring manager for the publisher's email address to say thanks) - which you should always do too!
The hiring manager asked the usual interview questions (what I was looking to gain from the internship, where I wanted to go in publishing, etc.) as well as some clarifying questions about my cover letter and CV. The supervisor asked more broad questions about publishing in general and what I thought certain aspects of publishing were. After the interviews, an email was sent to me (several days after the last interview) saying I was accepted as an intern with the terms of agreement . I just had to formally agree and set a starting date.