This app was mentioned in 2 comments, with an average of 0.50 upvotes
I recommend Shogi Wars. It's the most popular shogi app in Japan. And you can play english version. If you and your friend registers as "friend" each other, you can play online in "Game with friend" mode.
iPhone: https://itunes.apple.com/us/app/shogi-wars/id496801169/
Android: https://play.google.com/store/apps/details?id=jp.heroz.android.shogiwars&hl=en