You need to run a Query, as explained here in the documentation of the REST API.

Since you want to query all the messages sub-collections of different user documents, you need to "simulate" a Collection Group Query in your StructuredQuery. The way to do that is to set the allDescendants element to true in the CollectionSelector.

So, issuing a POST HTTP Request on the following URL will do the trick:

  var URL = "https://firestore.googleapis.com/v1/projects/<your-project-id>/databases/(default)/documents:runQuery";

The body of the POST Request shall contain:

    "structuredQuery": {
        "from": [{
            "collectionId": "messages",
            "allDescendants": true
        }],

        "where": {
            "fieldFilter": {
                "field": {
                    "fieldPath": "sent"
                },
                "op": "EQUAL",
                "value": {
                    "stringValue": "true",
                }
            }
        }
    }

Note that you need to add a single field index to your Firestore DB, as follows:

Note also that, if your field sent is of type Boolean (and not String as shown in your question), you need to use a booleanValue element in your Value JSON element.