If these are users accessing a hosted database at the same time, they'll each get their own "virtual copy" of the global field and thus won't see each other's data in this field. In a hosted database, each client's edits of a global field are kept separate and their edits are not retained, the next time they open the file, they'll see the original "default" value of the field.
Any use you make of the field as developer in single user mode, will persist, so clean up after yourself by clearing the field before putting it up on the server. If this is a single user file or you don't want users to see text in this field each time they return to it in the same session, you can have your scripts either clear the field just before the user brings up the layout/window where you have the global field or you can clear it after performing the find.
I prefer to clear the field just before. That way, I can add a "modify find" button that takes the user back for a new find with the original criteria unmodified. If they click the "find" button, the script clears the field before they see it.