You can visit other players' Home Islands!

Just tap on their name in any of the global rankings panels, and then tap on 'Visit'. You must be level 7 to unlock this feature. When you visit a player's Home Island, you can check out their decorations and if they've asked for help with their Helicopter Order you can help them. When you're done, you can tap on the home button in the bottom right to return home.

You cannot stop a player from visiting your home.