When a loosing defender of an intrigue challenge discards cards at random from his or her hand, do they get to choose what they discard?

The wording in the rules seems vague. Like, is the use of the word random to do with me being able to choose whatever I want or I have to turn my cards face-down, shuffle them up then discard X cards?

You do not get to choose. Random means you do not select the card(s).

So what is the standard way people discard their cards? Face-down, shuffle them round and toss em out?

The standard method is to fan your cards out and let the winner of the challenge take one out of your hand.