No, there is no standard mechanism for self-descriptive JSON.
However, I ran across this proposal not long ago. I like the concept of self-descriptive JSON, but I am not a fan of this particular proposal. First, the proposal defines some new JSON Schema keywords to make the schema self descriptive. This is unnecessary. JSON Schema is already fully self descriptive without any of these additions. Next, the proposal describes a JSON structure with a mechanism for specifying the schema that describes the data. I think this structure is quite awkward. But to be fair, the simplicity of the JSON format makes this kind of thing difficult to express.
If I needed to associate a JSON document outside the context of HTTP, I would do something like what Clemens describes and store it as a tuple of JSON document and schema URL.