A white dog is running through the water at a beach
The white dog is with it's owner.
The dog doesn't necessarily have to be with it's owner, if it was running through the water of the beach it could be alone as well.
The dog may not be white in color, and not with its owner.
A dog may be with it's[sic] owner or not with its owner or not have an owner and a dog running gives no indication of which is the case.