You don’t need to sign out of Skype for Android; the app will run in the background without taking up any battery power so you don’t miss any Skype calls or IMs. Skype for Android uses Android’s push notification system to let you know when you receive any new messages or calls.
Note that group chats will not appear when the app is running in the background – only 1:1 IMs and Skype calls will “wake” the app and notify you. You can select to be notified on group chats, as well, but this will cause more strain on the battery.
When you close Skype, the app will run in the background as above. If you don’t wish to receive notifications of messages and Skype calls, you can sign out of the app.