Thats funny you say that, I have worked in an airport for many years. The exact quote from the airport police and administrator on this topic is "The airport is a public place, we can't stop the public from doing things that don't impede security". If it is owned by a government or a government agency (which most are) that means it is public.
I'm not sure about the "most airports are owned by a government or government agency" bit.
I suspect this varies considerably from country to country. So; where are you basing your perceptions on?
In the UK the situation is quite muddled.
LHR; once the flagship of BAA, is now owned by Spanish company Ferrovial, who also operate GLA plus two others
LGW is Gatwick Airport Limited, a wholly owned subsidiary of Ivy Holdco Limited, who might be part of Global Infrastructure partners (a US Corporation) that also owns EDI. However I cannot be sure because the ownership trail is confusing. Now why would that be?
MAN is run by (Manchester Airports Group) which also run STN & EMA. MAG itself is 65% owned by ten Manchester Borough Councils, plus an Australian Investment Fund. To the outsider, it looks as if Manchester Airport is effectively owned by Manchester Council. However, when you consider Stansted, 150 miles down the road, the truth is revealed as something different.
Luton (LTN) remains in municipal ownership, but in 1997, to fund an £80 million extension of the airport, the council issued a 30-year concession contract to a public private partnership consortium, London Luton Airport Operations Limited, a partnership of Airport Group International (AGI) and Barclays Private Equity. AGI was a specialist airport management and development company once owned by Lockheed Martin Corp. of USA.
If you look at France, Charles de Gaulle Airport, Orly Airport and Le Bourget Airport are all part of Group ADP, who own & operate a total of 34 airports all around the world.
Basically, everywhere I look, I see either Government investment (but at arm's length) or non-Government ownership.
Nothing to see here; move along please.