For me hatching can be a long process of finding someone who has a pet you want, and then finding someone who has a pet you want that also wants your pet, and I feel like it would become easier if you could communicate on the hatching screen. For example- have you ever wanted for the other person to go through their pets so you could see what else they had, so you go through yours and they don't get the hint? Or they flipped through their pets but too quickly and you want to ask about one they passed?
I feel like even a simple menu chat would make a major improvement on this. You could just make options such as "Can I hatch with your..." and have the selections of all the pets the other wizard has as the options to complete that sentence. And "Can you go through your other pets?" so they could flip through their other pets so you could see the stats. And responses to those, such as "Sorry, no", "Sorry, I don't have any other pets", and "I want you to hatch with this pet"
Also sorry in advance if there already is something like that and I'm missing it, please let me know as it would make my hatching experience much easier.