You should probably do something like this to keep one user per row
-- update first
UPDATE table SET being_displayed_to = @user_id WHERE being_displayed_to IS NULL LIMIT 1;
-- then select
SELECT ... FROM table WHERE being_displayed_to = @user_id
When the user is done with the row, reset this field to null again.
You will obviously also need to deal with users leaving the row editing page through some other means than submitting the page to your server.
I believe you could deal with this by setting an event handler for window.onunload. But there is still no guarantee that this event will fire. Even if you can require javascript to access the site, a user could still disable javascript after acquiring a row, or the browser could crash. So you should also set a timestamp for when the row was acquired, and then clean set being_displayed_to = null after a certain time has passed.
Aside from this, I'd also recommend not using upper case for identifiers, at least if using mysql on a windows OS, since that is case insensitive. So if you move to another OS (or possibly another DB on windows), you risk introducing errors when the system becomes case sensitive.