Do I have relationships correct?

I am writing a RESTful chatbot, I have three tables, one for a person, one for chat and one for chatMessage.

A chat is comprised of chatMessages A chatMessage belongs to a chat and has a person (who authored the message)

see this gist for my models and migrations...

https://gist.github.com/matgargano/e4790425451aebc8fe105fca525c556d

Assuming the above is correct... if I want to get all of the chat messages from a chat I have an endpoint http://chatservice.dev/api/chat/1 which performs this query:

Chat::find(1)->chatMessages()->getResults(); and returns an array of objects like so:

{
    "data": [
        {
            "id": 1,
            "chat_id": 1,
            "person_id": 1,
            "message": "Ad alias iusto ea ut ipsum rerum.",
            "created_at": "2016-12-04 17:35:11",
            "updated_at": "2016-12-04 17:35:11"
        },
        {
            "id": 2,
            "chat_id": 1,
            "person_id": 2,
            "message": "Unde odio enim beatae et labore consequuntur.",
            "created_at": "2016-12-04 17:36:44",
            "updated_at": "2016-12-04 17:36:44"
        },
        {
            "id": 3,
            "chat_id": 1,
            "person_id": 2,
            "message": "Quod dolorum aut ut pariatur hic ducimus vero.",
            "created_at": "2016-12-04 17:36:53",
            "updated_at": "2016-12-04 17:36:53"

        }
        
    ],
    "code": 200
}

Two questions:

1. Have I got the relationships set up correctly in my models (see gist)
2. Is there any way that I can run an eloquent get and also pull in the person data from the person ID, so it would return something like:

{
    "data": [
        {
            "id": 1,
            "chat_id": 1,
            "person": {
                              "id": 1,
                              "name": "Jane Doe",
                         },
            "message": "Ad alias iusto ea ut ipsum rerum.",
            "created_at": "2016-12-04 17:35:11",
            "updated_at": "2016-12-04 17:35:11"
        },
        {
            "id": 2,
            "chat_id": 1,
            "person": {
                              "id": 2,
                              "name": "John Doe",
                         },
            "message": "Unde odio enim beatae et labore consequuntur.",
            "created_at": "2016-12-04 17:36:44",
            "updated_at": "2016-12-04 17:36:44"
        },
        {
            "id": 3,
            "chat_id": 1,
            "person": {
                              "id": 2,
                              "name": "John Doe",
                         },
            "message": "Quod dolorum aut ut pariatur hic ducimus vero.",
            "created_at": "2016-12-04 17:36:53",
            "updated_at": "2016-12-04 17:36:53"

        }
        
    ],
    "code": 200
}

instead

Thanks for any help at all.