"The people eat the bear."
Translation:Folket spiser bjørnen.
Actually while this sentence might look simple it is a really delicate thing to express in Danish.
First of all lets talk the 'official' sugested by Duolingo Danish translation: "Folket spiser bjørnen.". In this case 'Folket' means people as in "A body of persons sharing a common religion, culture, or language" and not as in "some group of humans". Wish is one correct interpretation of the English sentence.
An alternative would be "Folk spiser bjørnen.". In this case it "Folk" would refer to people in general, so more like "People eat the bear.". Or the people (in general) at this location/at this event, ""The people here eat the bear." Though in a very non-specific way.
You suggestion "Personerne spiser bjørnen." Translate to: These people eat the bear. Or actually more like "The persons eat the bear". And while 'persons' is wrong English 'personerne' is technically correct Danish though I don't think I have ever heard any body use that conjugation for any purpose, as it sounds wrong.
There might be other ways of putting it that would correctly convey the English meaning though these are the option that I could think of. All in all I think both the English and the Danish sentences are off from what one might normally say, and they are therefore hard to talk meaningfully about.