How can I share my Co-op Show private room number with other players?

Tap "Create Room" to display the Room ID and create a Private Room. Share the Room ID with the friends you want to play with.
Tap "Copy" to copy the Room ID to your device's memory. It can then be pasted into a text box.

If you're going to be using another app to share your Room ID with friends, note that you will lose connection if you pause the app after entering the room. Create your room after sharing the Room ID with your friends.