There are several ways to complete the sentence "A lot of local specialities..." based on the original statement, each with slightly different nuances:

1. Focusing on the sellers:

• A lot of local specialities are sold by local people at the open-air market. (This directly reflects the original sentence.)
• Local people offer a wide variety of local specialities at the open-air market. (This emphasizes the abundance and variety.)

2. Focusing on the products:

• The open-air market boasts a lot of local specialities sold by local people. (This emphasizes the market as a place to find them.)
• You can find a lot of local specialities, all sold by local people, at the open-air market. (This highlights the connection between the products and their origin.)

3. Combining both aspects:

• At the open-air market, visitors can discover a vast array of local specialities, offered directly by the people who make them. (This gives a more complete picture of the market experience.)

Choose the option that best suits the information you want to emphasize in your writing.