Must all property cards be where any player can see which one's I have?
Yes, the title deed cards must be visible. From the rules:
BUYING PROPERTY… Whenever you land on an unowned property you may buy that property from the Bank at its printed price. You receive the Title Deed card showing ownership; place it face up in front of you.
The official rules to Monopoly do not give any instructions on honesty, or on what information is open and which is hidden. Ownership of property is trackable (i.e. if you have a good memory you should know who owns which properties), so in that sense there is not much point to keeping it hidden other than being sneaky.
In fact, the closest the rules get to telling you about honesty is the fact that if you land on a property owned by another player, that player must remember to ask you for rent or they don't get it.