Most restaurants in Antigua that claim to offer local cuisine may or may not actually do so. If they do, it's still presented to impress a visitor, thereby taking away the actual authenticity of the dish. If you're a foodie, who's really interested in what and where Antiguan residents eat, I will give you a general idea.
The national dish of Antigua is Fungee and Pepperpot, but other dishes such as salfish and ducana and very much local dishes as well. The best way to sample all of the local cusines of the island would be at a food fair, but there are only a few of these occasions, scattered throughout the year.
Fungee is a dish made mostly of cornmeal and pepperpot is a stewed meat dish featuring spinach and many different types of meat such as chicken, pork and lamb. In order for a tourist to get a taste of these dishes, he/she would have to do some research prior to arriving, or ask a local for advice.
Small cafes and eateries may have local cuisne available Monday to Friday. These locations are definitely not first class restaurants, but the food is unbelievable. Below, I've listed two places where local cusine can be found in St. John's:
Hemingway's Caribbean Cafe: Located on St. Mary's St. on the second floor of an authentic old West Indian house, serves local creole cuisine Mondays to Saturdays, 8:30am to 11pm. $-$$
Island B-Hive: Located in Heritage Quay, local dishes are served for lunch. $-$$
Every weekend, there are small shops and booths which cook local dishes for their neighbourhood/village. These booths are somewat like roadside food stops, and the dishes are cooked by residents. Some of these booths are actually tents pitched up on a street corner, with tables laden with pots. Inside these pots may be cooked crabs, ducana, saltfish, pepperpot, creole fish, coconut dumplings and seasoned rice. The prices are definitely affordable (around USD $5) and the servings are large.
My biggest tip to find local food in Antigua is this: Where there is a large crowd of locals at a small booth or pickup buying food, ask what's on the menu. More often than not, it's the real deal.