I'm guessing the governments of the Federation Planets are all independent (in other words, Vulcan has a Vulcan government, and Earth has an Earth government, etc.) So while we've seen the "Federation President" in movies, who's running earth?
There is a basic idea that Earth is a sort of "utopian" society, but is it run by a ruling oligarchy? Socialist society? In none of the works that I've seen do I recall any mention of earth people of the 23rd century voting, but that's not to say it wasn't there, or it's not in some of the works I'm not familiar with.
There's a Sisko quote from Paradise Lost
where he says "Overthrowing a legitimately elected President and giving Starfleet direct control over the government? Sounds like a dictatorship to me.". This seems to indicate the Federation President is elected, but this could still be an 'election' among members of the Federation Council as opposed to a general election and popular vote. Still, I lean towards the former.
The Federation government is pretty obviously headquartered on Earth, but this is analogous to the Federal government of the U.S. being headquartered in one of its constituent states rather than the District, as it is now. One could argue this might be a better arrangement (as it does seem the Federation President has a tendency to meddle in what would otherwise seem to be Earth-specific affairs), but nonetheless this does seem to be the model we have.
I imagine there is a multi-level government on Earth that includes leaders for each town, each region, each country (who might still be called 'Presidents' of their respective countries, i.e. there might still be a POTUS) under the leader of United Earth, who might also be called the President (of Earth) and democratically elected. We've never seen him or her, but this person may well be responsible for the day to day running of Earth.
The Federation President has, from time to time, apparently exercised some kind of direct rule on Earth, such as during the Whale Probe incident, and during the worldwide power outage in Paradise Lost, this may be just for unusual emergency situations rather than something that occurs day to day. As the base for the Federation government, I'm sure Earth's government works with the POTUFP more closely than many other world governments do, but nonetheless I'd like to think there's some kind of separation.
The alternative would be to assume that all of Earth had somehow been declared as a 'federal district' and was ruled directly by the Federation, whilst other worlds maintained their own governments, but this seems somewhat unacceptable to me. Why should a bunch of aliens rule over Earth when they are supposed to be nominally representing their own planets to the federal government?
In terms of the 'invisibility' of the Earth government in the 24thC, this is really to be expected. In a world where crime is practically nil, money does not exist (or at least isn't used regularly), everyone has a replicator or access to replicated goods, everyone has artificially intelligent computers at their beckon call to provide answers, solutions and teach their kids, one imagines the role of the government would be reduced to an invisible overseer rather than what we have in Western countries today.
Their responsibilities would probably mostly include: Energy production and a worldwide grid of power distribution, organising transportation (of people and parcels) via the transporter system or those tube trains we see criss-crossing Paris and San Francisco, maintaining a nominal police force (crime hasn't been completely eradicated, but why steal when you can have anything you like, and forensics are a fine art form?), providing access to information (a computer for all) and replicators (if there's not one in every home, one imagines there's an industrial replicator at the end of every street or where a corner shop or supermarket would be in the present day), and health (access to hospitals where doctors practice medicine because they want to, not for a salary). Educational institutions may also be organised by the government but equally may be private societies organised ad-hoc by groups of individuals: schools seem to be, even if colleges and universities work like they do currently. Many children may be homeschooled by the computer...
One imagines that with this vastly reduced bureaucracy there'd be no need for taxes (what would you tax anyway in a society with no money?) or all of the regulatory machina we have in our government today.
People in general do things because they want
to: a chef opens a restaurant and cooks for people because that's what he enjoys, not because he's looking to make money, I doubt the patrons pay him for the food, instead he earns kudos via good reviews and happy customers, and that's his 'success' as a restaurateur. A Doctor doesn't practice medicine for a monthly salary, but because he wants to heal people. He signs on at a hopsital down the road because that's his life calling, not for the 200k/year salary. Likewise other professions, including Starfleet, who seem to be people mostly interested in exploration or 'frontier' science and medicine.
Whilst this seems quite socialistic, it's not socialistic in the same sense as socialism today - government is also very much reduced, so it's actually something that could appeal to people from any side of the political spectrum. What we're actually seeing is a 'post-scarcity' society, which is a fundamentally different form of society that cannot exist without ST's magic technologies, but seems like a natural way for things to go once anyone can have anything they want for free.