Many public spaces in cities around the world are actually owned by private money. This is true for Granary Square in London - one of the biggest open-air spaces in whole metropolitan Europe. Signs at entrances discourage misbehaviour by reminding visitors who sets the rules.
The litmus test for a public and democratic space is a political protest. Londoners can no longer protest in front of their City Hall, after 13-acre "public" estate was bought by a private company.
King of privately owned public spaces is the proposed Garden Bridge in central London. Marketed as a "gift" to all Londoners this private garden masquerading as a public bridge prohibits cycling, skating, playing instruments, protests, sleeping, exercising and releasing balloons. The bridge will have 24/7 private security force with powers to stop, search and confiscate illegal items (such as kites). Groups of 8 and more will not be allowed onto the bridge without an official permit in fears of protests. The bridge owners will not guarantee public right of way and will close the bridge every night as well as 12 full days every year for corporate parties. By the way, the bridge is constructed with public money.