As has been stated, there's no private messaging facility on AB.
The usual way that members get in touch with each other is by posting a message, giving a 'throwaway' email address, on a recent thread of the member they'd like to contact. (In VERY special circumstances the Ed might pass on a message to someone. A couple of members have contacted me that way - but it has to something genuinely exceptional that creates the request because the Ed clearly has far more important things to do than operating a messaging service).