I think it's a good idea that when someone order a Free Push, the mail "Dynadot Orders Order Received (order xxxxxx) (account xxxxxx)" is send to both the one who pushed the domain and the one who is going to recieve it.. so, no one gets nervous ;)
Both parties do receive an email. In a domain push, the current owner receives the "Order Received" email and the recipient receives a "Change Account Notification" email.