Are there any privately owned large commercial airports?
Building an airport is a very expensive endeavor and usually involves government subsidies or is completely done by the local government. In recent decades, most government owned airports in Europe have however been transformed into companies that both own and operate the airport, usually with the government as initial owner of this company. Since this privatization, some governments have sold their shares of the company resulting in completely privately owned airports.

Going through the 20 busiest airports in Europe from Wikipedia's list, the following airports are completely or by majority privately owned:


London Heathrow Airport owned by Heathrow Airport Holdings
London Gatwick Airport with a majority ownership by GIP
Rome Leonardo da Vinci-Fiumicino owned by Aeroporti di Roma (3.8% government owned)
Zürich Airport owned by Flughafen Zürich AG (38.38% government owned)
Copenhagen Airport owned by Københavns Lufthavne (39.2% government owned)
Lisbon Airport owned by Vinci SA, although operated by the government


The following airports are run by a private company, but they are majority government owned:


Paris Charles de Gaulle Airport and Orly Airport owned by Groupe ADP (50.6% state owned)
Amsterdam Airport Schiphol owned by Royal Schiphol Group (92% government owned)
Frankfurt Airport owned by Fraport (51.47% government owned)
Madrid-Barajas, Barcelona-El Prat and Palma de Majorca owned by Aena S.A. (51% owned by ENAIRE, which is a public corporate entity)
Munich Airport owned by Flughafen München GmbH (100% government owned)
Oslo Gardermoen owned by Avinor (100% state owned)
Manchester Airport owned by Manchester Airport Holdings (64.5% government owned)


The following airports are directly owned by the local governments:


Istanbul Atatürk Airport
Dublin Airport


I am not sure who exactly owns Sheremetyevo and Domodedovo in Moscow.