How would you know if you're recovering from a spinal injury?

Your recovery. It depends on the defecits that you had after the injury, if you are recovering these functions, then you are recovering from the injury.
You feel better and. If just an injury causing pain, the pain will subside. If there was a neurologic injury leading to some deficit, there will be signs of that deficit (s) returning such as feeling or strength or movement. You physician should be able to guide you on the prognosis based on the type of injury and direct the appropriate rehabilitation.