Yes. There are two parts to the answer. First, when a person has performed an act of courage, bravery, valour or merit, it is the chain of command that decides that the person should be recognized. That is a very important effect.
When the chain of command, through the system, initiates the process for obtaining government approvals, the government accepts the recommendations by the Canadian Forces, and the medal, decoration or order will be issued, very probably by the Governor General, at an appropriate location. At the time of presentation, the medal, and we're talking about a generic medal, is presented in good faith. It really is a tangible gesture to reward individuals and recognize what they did. It is precisely a symbolic sign of recognition for what they did.
So in terms of ownership, awarding a medal means it has been given to the individual.