I’ve never worked for a federal client, but numerous times for state agencies and almost as many times for city or county government. The state agencies included the Administrative Office of the Courts, School Boards, projects funded by Community Development Block Grants, and regional Agencies like planning, water and sewer districts. Finally there are the Counties and Cities. These tend to be the least bureaucratic and most like private clients except for that one thing that is common to all public clients - procurement regulations. These regulations are very similar in intent, which is preventing misuse of public funds by closing loopholes. The big difference for the architect is that several additional tasks are required and there isn't a lot of leeway in how you do them.
This link takes you to an article on the bidding process.
Besides the procurement regs there are a few other issues that come up; they are listed here by phase.
Politics can be a wild card in the process. It is wise to remember the naval advice: "Loose lips sink ships."