I think that they could be replaced, but there would be some issues without a governing authority, such as child slavery, black markets, slavery in general, etc. In some ways, P2P government wouldn't be bad. However, thing like law enforcement would turn into bounty jobs, so there would be a sudden increase in either bounty hunters or mercenaries.
For other government functions, however, it could be good or bad. No president = anarchy, so you see what the issue becomes pretty quickly...
Those are still issues with a governing authority; in fact, black markets are more popular than ever, there's several based on bitcoin going on right now. In fact, black markets exist only because government attempts to force people to not buy certain items; without them, there would be no problem, all "shady" businesses come to the light and are far less dangerous that way. Furthermore, slavery was only profitable because government socialized the cost of capturing slaves; when governments stopped doing this, slavery ended very quickly. While slavery is still a problem in some areas, the fact that it's still a problem, despite all this government entering our lives, means that it's going to continue to be a problem with or without government.
Law enforcement is a government-created institution; in its place would be defense agencies which literally do serve and protect, unlike the ones we have now. And it's false that no president = anarchy, it's no state = anarchy; you can still have a president in an anarchic system, you just wouldn't force him on people who don't want him as their president. Consider the following video for more information on how it works:
https://www.youtube.com/watch?v=muHg86Mys7INo. It cannot.
National security cannot be done in a peer to peer fashion. If this were to be even attempted the US and the world would be doomed.
Nationalism is inefficient; you can have local security, and militias, and fancy defensive weaponry very easily if people would stop forcing others to live under one federal roof. You can't defend a whole nation without the state, but it's because of the state that people are subjected to a nation in the first place. Furthermore, there is nothing a terrorist or another nation would want with a non-nation; there's nothing to conquer, no government to influence with terror, and no infrastructure to take over if an invasion occurred: the only thing that makes empire profitable is the fact that empires exist at all.