In Monopoly, how do people determine who is the owner of a property?
When I've played Monopoly, the owner of property always kept the property card ("title deed") on the table in front of them, visible to all players.

The bank should also keep the unsold property cards visible to everyone; that way, you won't need to call attention to yourself if you land on an owned property.  That is, you shouldn't keep track of who owns property in order to pay rent; it's up to the owner to keep track of what property they own and ask for the rent.  "The owner may not collect the rent if he/she fails to ask for it before the second player following throws the dice."

An example of how each player can organize their properties, as well as their money, can be seen in the image below: