Verify JWT Claims in WebAPI

Given the Thinktecture AuthenticationConfiguration below:

var authConfig = new AuthenticationConfiguration
{
    EnableSessionToken = true,
    SendWwwAuthenticateResponseHeaders = true,
    RequireSsl = false,
    ClaimsAuthenticationManager = new ClaimsTransformation(),
    SessionToken = new SessionTokenConfiguration
    {
        EndpointAddress = "/api/token",
        SigningKey = CryptoRandom.CreateRandomKey(32),
        DefaultTokenLifetime = new TimeSpan(1, 0, 0)
    }
};

It would return an example JWT of eyJ0eXAiOiJKV1QiLCJhbGciOiJIUzI1NiJ9.eyJpc3MiOiJzZXNzaW9uIGlzc3VlciIsImF1ZCI6Imh0dHA6Ly9zZXNzaW9uLnR0IiwibmJmIjoxNDIwMzk2ODgyLCJleHAiOjE0MjA0MDA0ODIsInVuaXF1ZV9uYW1lIjoicGFzcyIsImF1dGhtZXRob2QiOiJodHRwOi8vc2NoZW1hcy5taWNyb3NvZnQuY29tL3dzLzIwMDgvMDYvaWRlbnRpdHkvYXV0aGVudGljYXRpb25tZXRob2QvcGFzc3dvcmQiLCJhdXRoX3RpbWUiOiIyMDE1LTAxLTA0VDE4OjQxOjA0LjAxOVoiLCJyb2xlIjoiVmVyaWZpZWQifQ.h7curaLrqkMT4Btg-AAoEpNYqUIYNQA_y-eUdEwQBqs

Which is:

{
    "alg": "HS256", 
    "typ": "JWT"
}

{
    "unique_name": "pass", 
    "aud": "http://session.tt", 
    "iss": "session issuer", 
    "authmethod": "http://schemas.microsoft.com/ws/2008/06/identity/authenticationmethod/password", 
    "role": "Verified", 
    "exp": 1420400482, 
    "auth_time": "2015-01-04T18:41:04.019Z", 
    "nbf": 1420396882
}

How would I verify that the JWT was issued from a trusted machine, can we use a symmetric key for the private signing key and the same key on the remote machine to verify against?

How could I wire up the WebAPI so that it automatically does this for us (assuming the AuthenticationConfiguration is on a different machine dedicated to account security api).

You can use a shared symmetric key or a private key to sign the JWT and that use that same symmetric key or respectively the associated public key to verify it.

The algorithm in use for this JWT (HS256) suggests that a shared symmetric key was used so you need to know that symmetric key at the receiving end in order to verify the JWT.