Good article on "proprio": http://italian.about.com/od/grammar/a/italian-possessive-adjectives.htm
RobDB's answer is correct! More examples are shown at this link (scroll down some to get to the section on "proprio") http://italian.about.com/od/grammar/a/italian-possessive-adjectives.htm It also shows you all the forms (masculine/feminine, singular/plural).
If Duolingo insist on "their own" when teaching Italian speakers English, they will be teaching them bad English.
In English saying the same thing twice in different ways within the same sentence is known as a tautology, and is considered to be bad style.
English use of the possessive "their" implies that it is their own. We would not use their if it belonged to someone else. (Probably just "They eat the cake".) Presumably Italian needs the "propria" to make that distinction, however, we don't.
To say that they eat their own cake is to emphasize the ownership. It would be used in the context of perhaps arguing about whose cake is being eaten by "them". It's used quite a bit in English.
The initial statement in such an exchange might be: "They're eating their cake." Followed by, for instance: "No, it's mine." Followed by: "It's their own cake." (With tonal emphasis on the word "own.") So, "la loro torta" followed by subsequent argument "la propria torta." None of this is important to argue about, except for anyone also learning English.
When the cake they eat is either their own cake (they own it, or they made it) of is the correct cake for them (chocolate, sugar free, et al) proprio makes this clear
If they eat someone else's cake ( that someone else made or that they shouldn't have eaten, eg) proprio would be wrong
So, in all the other forms like you (tua) i (mia) we (nostra) it just has an 'a' at the end, but here for the 'their own' it has 'ia' propria, is that just how it is
It is obviously a lesson for us to understand how Italians use 'proprio' no matter how WE translate it. I answered ' They eat their cake ' and was marked as wrong. Reading all the comments below helped me realise that, no matter how we translate it in english, we have to play the Duo game to show that we understand, it would be the same in a face-to-face lesson with a teacher.
When you use 'theirs' you must leave out the subject or move it to the front of the sentence. . You would say "they eat theirs" or "they eat their cake". Another example would be if somebody asked the question "Who's dog is that?" You would reply either "it is their dog" or "it is theirs". An alternative reply could be "the dog is theirs" which avoids the subject following "theirs" by moving it to the front of the sentence.