The regulations on space flight are currently being rewritten, even as we speak, as mentioned. I haven’t studied the proposed regs in detail.
At present, below 60,000 feet altitude the shuttle falls under air traffic control and a number of the current FAA regs - but the shuttle is also a glider when on descent, which gives it several advantages (like right of way over everyone else who has power) and there’s only one flying at a time, so reserving large blocks of airspace for it is practical. This won’t work when you have a number of private spacecraft in the air at once, and if any of them are capable of powered flight on the descent different rules will apply.
You do not need permission from a country in order to overfly it at orbital hieghts.
As to what form of licensing you’ll need to fly it… well, the regs are being written up for licensing the spacecraft. I think the US X-prize contenders are currently working under the regs for experimental aircraft in general, with appropriate waivers/permissions from their local Flight Standards District Office for test flights requiring deviations from the current rules as written. As for the pilots, I think up until now astronaut pilots have also been licensed airplane pilots and I don’t think that will change. What sort of additional training and certification you will need beyond just the basic license is somewhat up in the air. Up until now, NASA has trained people as they see fit and presumably have some sort of guidelines for their own use, but when private space travel occurs there is, at present, no requirements I am aware of and at least in theory such a pilot would not need to have gone through NASA’s program.
Rutan and company, who are one of the X-prize contenders, have a number of pilots experienced in experimental aviation and record-setting, but there’s no certification of “experimental pilot”. If you want to design, build, and fly your own unique design around the world all you need is a private pilot license (and very much you’d do so at your own risk). Since I am not an expert on private space travel I could be wrong, but my reading of the regs says that, at least in theory, all you would need to pilot your private spacecraft into orbit today is a private pilot license with an instrument rating, because you’d need to traverse Class A airspace, and those are the minimum requirements for doing so. After the new “space regs” are finalized that may change.
As for “government permission” - that depends on how you define the term. If you consider an air traffic control reservation “government permission” the answer is yes. But that’s not a huge deal, it’s basically making a phone call and given a little bit of information on what your magic carpet looks like, where you are, where you’re going, and when you want to leave. I find it less onerous than making an airline reservation. If you mean do you have to fill out forms in triplicate to submit in advance, pays multiple fees, and jump through a bunch of hoops I don’t know - and neither does anyone else right now because the rules are still being written.
Ask again in six months and there may be a more definite answer.