Lobster — Food safety facts
Safe to share · Dogs · full breakdown on the linked food page
Plain cooked lobster meat is safe. Never give the shell or drawn butter.
Short answer
A few bites of plain lobster meat are fine, but half a tail is a lot for most dogs. That much rich seafood can cause an upset stomach or pancreatitis, especially if it was cooked in butter or oil. Offer your dog plenty of water, watch for vomiting or diarrhea, and call your own vet if anything seems off.