I technically interviewed for this job twice. I turned 1 store down and accepted another stores offer. It boiled down to getting full time status for me. Oh, worth mentioning, Publix likes to make "full timers" shift leads basically. I had no idea I was accepting a leadership roll, so that was interesting. Now if you picked up the "full time" status meaning you're a "shift lead", that means Publix generally only hires part time staff, you'd be correct in your assumption.
So this is how my interview process started. You get a phone call asking if you would be interested in scheduling an interview. Once you pick a day and arrive, you will inform Customer Service (CS) that you are their for an interview for a job interview. They will call you back once they are ready for you to interview. Anyways, for me, both of my interviews were done by the Store Manager. They start off with standard questions and in the end both of my interviews went well and I was offered a job for both. I'd say the interviews stopped about half way and I ended up having a general discussion with both managers.
My advice to anyone interviewing for this job, come ready with questions about the deli. While Publix does have its standards for work, every kitchen runs their deli a little bit differently. So just keep that in mind, and good luck!