I don't know about other pet hatchers, but I would like to be able to save an image of the parents of my pets to see what Talents the other person's pet had. You can't save before the hatch because the other person might change their mind.
So how about after a pet hatch, we get a screen that holds the pets. Going from memory, I believe we get a screen declaring the other person has left.
-- Edited -- Other people might be responding to the original post, terribly sorry for that.