In one of the last scenes of Arrival, Louise boards the ship on her own and communicates with the alien Costello simply by speaking English. In the previous scenes, she uses a computer monitor to display the symbols to the aliens. How was she able to do this, and why didn't they do so before?
Right from the beginning, they spoke English with the heptapods. Remember how they taught them the words Louise and Human?
They used the heptapod language to make sure they were learning it correctly, and because they could make some more complex sentences with a higher probability of understanding from the other party. However, the heptapods were also learning English (or, as is my opinion, they knew it all along) and could understand them to some degree.
Louise and Ian specifically discuss the theory that submerging yourself in a language is the fast way to learn it.
Louise already had a rudimentary understanding of their language, but she still relied on the application to parse the symbols for her.
When she visits Costello, she doesn't have her translator and thus is fully submerged in the language. This is the first time where she does not actively rely on her translator - before this, she has always been shown to use the translator. After this visit, everything clicks for her, because she had left behind the crutch (the translator) that kept her from truly connecting with the language.
The important part here is that at the time of Louise's surprise visit, Costello was also being submerged in the English language, after having been initially exposed to English at a slower pace (at the same time as when Louise was learning their language during all of their previous visits). Every word the humans and heptapods discussed can obviously be learned by both sides (since they cannot learn it without the other party teaching them).
So it makes sense that Louise and Costello's experiences are (coincidentally) mirrored, as they are both being submerged in the other's language.