It doesn't count owners or co-owners, it only counts a person ONCE a day, a day being server up, so if someone enters your house at 7am in the morning and server up is at 6am in the morning then that character can come and go for the rest of the day and it doesn't count them again. If they log on a different character it counts as another visit. If they come the next day it counts them again. You don't need a door to open, but the person must enter the house for them to count (ie step on the stairs to load the house). Not sure if the house has to be public as all my houses have always been public, although my castles and such the doors are not accessible.
Tested all this out when I was running an event from my rune library, is why I worked out exactly what generated a 'visit' the event was to 'guess' how many visits the library had in a month, so the amount of existing visits was listed on the bulletin board at the start, people could 'guess' by writing their answers on the board, and at the end of the month a screen shot was taken of the house gump, posted on the shard forum in the thread on the event, and the new 'amount' was posted on the board, the person closest to the number got the prize.