You'll be unlikely to find a quote for this, so I'll draw upon personal experience as a player and referee.
First, let's clarify the referee's role in stopping play for injuries.
Law 5 - The Referee, Section 3 - Powers and Duties:
- allows play to continue until the ball is out of play if a player is only slightly injured
- stops play if a player is seriously injured and ensures that the player is removed from the field of play ...
So if the referee hasn't stopped play, they've decided that the player is only slightly injured.
Ettiquette here really depends what is happening when the injury occurs or is noticed. If the other team are in the middle of a promising attack, there is not really any expectation that they put the ball out of play. They've done nothing illegal - so it's not fair that they're punished for fair play, and even if the ball is returned to them, it's not going to restore the strong position they were in.
However, if the other team is simply knocking the ball around the back, or not really doing anything with it, the team of the injured player will generally expect that the ball will be put out of play. In return, the team of the injured player will give the ball back, and the other team will be effectively in the same position they were in before the ball was put out of play.