I'm going to step in and train another persons trainee
If you step in and train them is that fair?
Of Course it's unfair, I don't care getting people mad at me.
You want to help which is great, but do you really want to risk people getting mad? If you aren't asked to do something, let them do it themself.
Nah, I will show the them how everything is done.
You're making that person feel incompetent by stepping in their job. People at your company are not going to appreciate it, in fact it's rude. If you have any suggestions for the training you're welcome to make them but you're not supposed to take over that person's job.