Only the General and the Brigadiers can send 1-on-1 messages to alliance members.To send a message, tap on an alliance member’s tag in “My Alliance” or “Alliance Leaderboards”. This will surface a bunch of option for you to select. To send message, tap on “Message” option.
This will open up a page where you can type the message you want to send. Tap on send when you are done typing the message.
When a message is send to a player, a cool down timer for messages is applied to that player. The General and the Brigadiers cannot send a member any message with a cool down timer, until the cool down timer is over.
The recipient of that message will receive it as a push notification.You can check any message you have received in your “Message” tab, accessed through the “Chat” HUD icon (below the alliance icon). 
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Empires & Allies Game GUIDE