I asked Fishing Cactus on their Tumblr about the Social Mode and this is what I have gotten:
"Hi! Social features are available at all times so strictly speaking, there is no ?social mode? or ?solo-mode?. At any time you are able to send your eggs to a friend. When you receive an egg, it goes in the hatchery machine and from there you are free to hatch it, discard it or send it to someone else.
Hope this clarifies it a bit!"
Don't believe me? Here's proof!

And don't say it's photoshopped because I suck at that.