"She likes that newspaper."
Translation:Ela gosta desse jornal.
Desse and deste is just a combination of the preposition de and the demonstrative pronouns esse and este. You should use esse (and therefore, desse) when the thing you're pointing or talking about is far from you and close to the person you're interacting. On the contrary, este (and deste) must be used when the subject is closer to you than it is from the other person. When the subject is far from both of you, use aquele (and daquele).
In the informal speech, people don't care much about when you use esse or este, so you can use them interchangeably, but that's not the case of aquele.
In a text, you use este, esse and aquele to refer to something that is near or far, depending on the context. For example:
João e Pedro brigavam no pátio. Este ficou com o olho roxo, enquanto aquele se feriu no joelho.
- este: Pedro
- aquele: João
If you are translating EN to PT, you know the newspaper is away from the speaker which is the important thing. The speaker doesn't say "this" newspaper so "este" is strictly wrong. If you were translating PT to EN, you get a different perspective of where the newspaper is, but you are right there is nothing to tell you where it is with respect to the girl and it really doesn't matter.
Whether Duolingo should accept your answer is another thing altogether and you need a native speaker to adjudicate.