I work for a church, and many times fill in for the wedding liason. A church is not a public building. It is considered private property because they are owned not by the state, but by (and this depends on the denomination) trustees, administrative council, conference etc. We can have people thrown off the property or out of the building at any time (not that we have ever or would ever out of the building, but we have had to have "campers" leave leave the wooded property before)
The church is usually open, however, not all rooms at all times. If there is something going on (class, wedding, funeral etc) and someone uninvited shows up, there are other rooms in the building they are invited to use.