Club Tortimer requires WiFi, yes. You're going to an online island with other players from around your country/the globe. Some items are exclusive only to that island, while others you can just find at your own.
Yes, pretty sure you need WiFi for the international island. It's like the normal island but different items are there and you can play with others from around the world. I find it really fun.
But beware some people go there and beg for bells or troll.
The club tortimer paper won't appear on online islands. If you have it on your private island, then visit with friends it will be replaced with something else.