Submitted application online. Heard back from a recruiter couple weeks later and talked on the phone for 45 minutes about working in the public sector. Talked to another recruiter few weeks later about an opportunity in Atlanta, followed by a phone interview with a specialist on the team. Onsite interview in Atlanta: case interview with a senior manager, behavioral interview with a partner, behavioral interview on the phone with senior manager of the team. Received a contingency offer a month later. The contingency offer was based on whether or not the project will get funding from the client, which turned out it did not. Invited to talk to another team since the project I was "hired" for did not get the funding. Never heard back.
Overall, everyone I talked to was nice and professional. Deloitte is a big organization and has a lot of opportunities, but in order to be successful there, you need to network, network, and network.
Practice case interviews and use STAR technique for the behavioral interview. For the public sector, the cases are not your typical business cases, but the concepts are similar (improving efficiency, reducing cost, etc.)
They seem to gravitate towards individuals with entrepreneur spirit.