By default, the annotations are stored in a local SQLite database. But you can also configure Prodigy to use PostgreSQL or MySQL (see here) or write your own fully custom database adapter (see here for an example).
Using the db-out method, you can export a dataset to a JSONL file. Each entry is represented as prettx straightforward JSON, so it should also be no problem to use the data for other processes outside of Prodigy.
prodigy db-out your_dataset > annotations.jsonl