When you dm a player (by clicking send a message on their profile) it gives you the option to send items, the other player has to send something back and you have to accept it before you can message with them again but thats how private trades work!
Also want to say that you don't get notified when people accept the trade. You need to go back and claim the items/wolves from the trade in the message.
I don't know if the initiator gets a notification, but the two trades I've added to, I didn't get notified when the initiator accepted it!