In most US states, you’re liable for the actions of your dog, period, regardless of signage. In a minority of states, you’re normally not liable unless (a) you know that your dog is dangerous, (b) you broke an animal control law, or (c) you were otherwise negligent in handling your dog.
If you know that your dog is aggressive or dangerous, many state and local authorities require you to take additional precautions to protect the public from your dog. The definition of “dangerous’’ varies by state. Signs may be required. Check with your attorney or local animal control agency for details.
The goal of most Beware of Dog signs, of course, is not just to comply with the law. Tehse signs serve both as a way both to warn unwary visitors that they may encounter an aggressive dog and, next, to deter unwanted visitors or potential burglers.