We have a hybrid permit system here. The state is in charge of permits/inspections [/i]unless[/i] a city decides it wants to do their own. State permits are very old fashioned (quadruplicate carbon form), but they are pre-numbered and must be purchased for $1/each. They are considered valid once they are deposited in the US Mail system. I often drop the permit in the mail on the way to the job. Minimum fee is only $20 for a state permit. State inspectors are at their phones for an hour and a half every M-F morning, and if I call then, they come out the same day, rough-in or final. No one has to be present for the inspection; all they care about is being able to get in somehow. We make arrangements by phone. It's a low-tech system, but it seems to work pretty well and is not overly expensive. The one downside is that when the inspector says he'll be out that day, it could be anytime, so if no one is home, there must be another way for them to get in. Some people refuse to let in an inspector if they aren't home, so sometimes inspections are hard to arrange. I'm sure not going to sit around someone's house all day waiting.
Cities are all over the board in how their permits work. Some permits I can pull online, sometimes I can fax in a request, sometimes in person, etc., but the permit must be issued by the AHJ before work can begin. Cities are very clear that dropping the paperwork in the mail may be OK for state permits, but not for city ones. City inspectors generally also don't need the EC to be present for an inspection.
Almost all of my work is done with state permits, and most of the rest tend to be in a jurisdiction where I can pull the permit online. I don't mark up the permit fee, but if I have to go in and get the permit, I add time to my labor charge.