Good question. I'd always felt it a little strange that although there were dozens of languages in Middle-earth, there were only two writing scripts(albeit used in different forms). It would make sense to think that the Easterlings adopted Cirth, learnt from the Northmen or the Dwarves of the East; and that the Haradrim used Tengwar, learnt from the Black Numenoreans. However I'd like to think that there were other scripts in the East.
However, the following quote implies that Sauron may have invented script of his own and taught it to his servants.
Quote:
It is fashioned in an elven-script of Eregion, for they have no letters in Mordor for such subtle work.
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Taking all this into consideration I would say that the different ethnic groups of the Haradrim and Easterlings(JRRT said that they were a mishmash of different diverse peoples) used multiple scripts.