Forget my last answer : it was a stupid one... I tried to answer quickly, but with tiredness, it does not give good results.
For me, your problem of performance comes from the "WHERE NOT EXISTS (query)" because your query is executed on each result of the outer join.
I tried to figure out how you can avoid this with your current database design, but I did not found any solution. Maybe someone on the forum will have an idea.
If not, what I can propose your is to arrange yourself so that, for each couple (message, user) of your database, you have a corresponding entry in message_property, so that the first solution I proposed you (with an inner join) will work. And with multi-column indexes, it should be fast.
To do so, you can use trigger mechanism on both the insertion of the message to create all message_property entries of that message, and on user insertion to create all message_properties of the user, so that you do not need to change anything outside your SQL design.
The disadvantages of this solution are that the insertion of a new message or of a new message will be slower, and that your database size will be greater, but it should solve the problem of fast determining all read or unread messages of a dedicated user.
No, that also will be treated as an INNER JOIN, because it kills tuples where prop is null. I need entries where prop IS NULL (hence the LEFT OUTER JOIN) because messages without an entry in message_property must be treated as unread, the same as messages with an entry in message_property where is_read=FALSE.