When you find eggs you have the option to gift them to friends. New eggs recieved cannot be gifted again, but duplicates can.
Basketing option where you will be presented a list of friends in which you can send the egg to. Most eggs can only be sent to one friend, but some allow you to send a single egg to multiple people. Note that each person which receives an egg counts against your daily limit of Hatchlings requests, regardless if you send each egg to one person only or you send one egg to multiple people.