As you said, "The teacher in Beirut" is a phrase, not a complete sentence. So it would be grammatically wrong had it been written alone. However, "The teacher is in Beirut" is a complete sentence.
However: In teaching Arabic (and all other languages) duolingo often gives us phrases to translate, not only complete sentences. If the exercise said "translate the complete sentence" I would know the answer, but I have never seen that anywhere in duolingo.
Does anyone have a better answer?
I think the period means it's supposed to be a complete sentence. I haven't noticed whether this course is consistent about that though.