MySQL query to select messages related to a user
Some thoughts:


You should consider using prepared statements.
Remove $chat_type as parameter, as it is not used
Consider stronger validation on $page_type() parameters(like that is it null or integer). Right now you can easily break the function
Don't use * selects.  They don not help the code reader understand what sort of data is being returned without having to look a database.  They also tend to cause more bandwidth consumption than is necessary and make your queries fragile.  What if you add a new audit field to the database table?  That could then break your object instantiation later in the code if an unexpected field shows up or cause you to go modify that object's class for some reason that is unrelated to what that object does.
Consider COALESCE instead of IFNULL for determining conditional values (this is more suitable function in that it does not have potentially hidden value typing behavior). This is probably more common as well amongst MySQL developer types.
Write your query in a more readable form (not a series of concatenations, drop unnecessary parenthesis, ne consistent around using backticks, and capitalization of reserved words, etc.)


For example:

SELECT
    direct_message.{field} AS {field_alias},
    {other fields you need} AS {field_aliases}, 
    COALESCE (direct_message_group.staff_id, 0) AS staff_id
FROM direct_message
LEFT JOIN direct_message_group
    ON direct_message_group.chat_id = direct_message.id
INNER JOIN direct_message_thread
    ON direct_message_thread.chat_id = direct_message.id
WHERE
    (
        direct_message.recipient_id = ?
        OR direct_message.creator_id = ? 
        OR (
            direct_message_group.staff_id = ?
            AND direct_message_group.active = '0'
        )
    )
    AND direct_message.school_id = ?
    AND direct_message_thread.school_id = direct_message.school_id
GROUP BY direct_message.id
ORDER BY direct_message_thread.inserted DESC
LIMIT ?, ?


Here I have applied named parameters instead of you string concatenation.


You query makes you relationship between tables seem unclear. On one hand you join direct_message to direct_message_thread on some sort of chat id value, yet you also put in a WHERE condition that school id's for these records on these tables must be equal.  Should this just be a part of the join condition as opposed to WHERE clause filter?
addslashes() may not be appropriate for escape purposes and it is not being used to escape $start (offset value).  I would suggest you either use prepared statements or an appropriate DB-connector-specific escape functionality. Also, if you know the values of your class methods calls are going to be integers, for example, do you really need to escape these?
You object instantiation for direct_message class seems weird.  Why not pass the row data upon instantiation rather than having to call separate _populate() method with the data?  Why start that method name with an underscore?  Consider UpperFirstCamelCase for all class/interface names instead of snake_case.