The second option is "easier on the tongue", sounds more casual and natural.

But both are correct. However there is a slight difference of meanings, that almost never matters, but in some specific situations may be.

Literally, the second option reads like "Do dogs at yours (your place, under your supervision, etc) eat apples?".

Typically, it is the same. If a dog lives in your house - then it most probably is your dog.

But imagine some shelter, or pets hotel, where both owner's owns dogs and temporary cared for dogs are staying. In this rather specific situation there might be a suspense that the place owner treats his dogs and dogs trusted to him differently. In such a situation, the 2nd option would imply all the dogs, while the 1st option would imply only owned dogs apart from dogs in custody.

Compare "Чем у тебя собак кормят?" and "А чем ты своих собак кормишь?" - "What meal is fed to the dogs at yours?" vs "What meal do you feed to your own dogs?"

Both are correct phrases. In casual speech they would most probably convey the same idea.

The second option is "easier on the tongue", sounds more casual and natural. Not much different, very close, but if out of context, i'd rather chosen second way to express myself.

However there is a slight difference in their meanings, that almost never matters, in casual chat, but in some specific situations may do.

Literally, the second option reads like "Do dogs at yours (your place, under your supervision, etc) eat apples?".

Typically, it is the same. If a dog lives in your house - then it most probably is your dog.

But imagine some shelter, or pets hotel, where both owner's own dogs and temporary cared for dogs are staying. In this rather specific situation there might be a suspense that the place owner treats his dogs and dogs trusted to him differently. In such a situation, the 2nd option would imply all the dogs, while the 1st option would imply only owned dogs apart from dogs in custody.

Compare "Чем у тебя собак кормят?" and "А чем ты своих собак кормишь?" - "What meal is fed to the dogs at yours?" vs "What meal do you feed to your own dogs?"

Also, when such a phrase arrives in the middle of a long discussion, it inherits the history and phraseology already established. Like, if we are both dog fans, and we just went for half an hour discussing "your dogs" vs "my dogs", then the 1st option would have more chances to be chosen. Because it would refer to the previously kind of coined "my dogs" and "your dogs" notions.