The next line should do the trick. This will give you all the owners where at least one of their dogs is on the local list.
owners.Where(owner => owner.Dogs.Any(dog => localDogs.Contains(dog)))
If you only need owners where all of their dogs are on the local list, use the following.
owners.Where(owner => owner.Dogs.All(dog => localDogs.Contains(dog)))
A Tiny Difference - Any() vs. All() .
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