I have successfully registered the twitter connector with my app and created a user. I'm able to send tweets and the test and info apis both work. Only when I try to retrieve messages the connector returns the following result:
{"Status":{"twitter":{"status":417,"info":"417 Connector returned error: 403 Forbidden"}},"messages":{"twitter":null}}
I send the request to /v2/message/retrieve with the following json body:
{"uri": "unified://twitter"}
The problem was that my twitter app didn't have DM permissions. I just saw Read & write and thought that would be enough.
If someone has the same problem, go to your app settings, change the permissions and regenerate your personal access token.