You can setup email notifications to get emails when jobs pass certain stages in Translation Manager.
You can setup notifications globally in Settings -> Translation Manager -> Notifications.
You can override the global settings on a set by set basis from the translation set configuration page.
Notifies when items are put into the pending queue.
Notifies when Translation Jobs are submitted/created by editors.
Notifies when a translation job is received from a translation connector.
Notifies when a translation job is approved in Umbraco.