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No, it is not necessary in this case. I suppose we have decided as a people to get rid of some redundant articles. BUT! It's never wrong to use it, when in doubt.
Also, I've noticed when these questions come up, that when there are two articles in a short sentence (like the duolingo sentences) it is safest to not skip the first article, if you're not sure. That one is usually doing a better job than the second one. This is an untested hypothesis, so if anyone does this, let me know if it works well. But r_i_l_e_y has a good point there, you should listen to him.
And Angola, and Mozambique, and East Timor, and Macau, and Goa, and Guinea-Bissau, and Cape Verde, and São Tomé e Príncipe, as well as Daman, and Diu, and Malacca in Malaysia...
Added to the 27 million all over Brazil:
Actually seus/suas is connected to the plural of keys (chaves) here, not the singular or plural of the 3rd person pronouns. So yes, it should be possible for the answer to be, "their keys."
Ela is the subject noun, while chaves is the object noun (possessed in this case) and the possessive is attached to the object independently of the subject here.
To avoid that ambiguity the sentence could be structured as, "Ela perde as chaves deles/delas." (She loses the keys of theirs). At least if "their keys" was what was meant.
With the 3rd person of "seu/sua" there is much ambiguity as it could be yours, his/hers, theirs, its, and even the plural yours). Well, that is one reason many of the Brazilians mix in the 2nd Person declensions of teu[s]/tua[s] because those only mean "yours" in the singular.