But there is a flip side that may be discussed less: what to do with the Really Really Good trainees.
The temptation to hold on to a good trainee is large. I've seen it, and tried to be aware when I've got such inclinations. A good trainee moves your lab forward. A good trainee challenges your thinking. On good days you see it right away. On bad days you struggle not to shut them down when they challenge you.
I've had some incredible trainees. It doesn't take much of a push to remember ones from even 25 years ago: their names are on some of my favorite papers. Papers I love because they aren't in the standard journals. They're in other journals because they had a bit of this and a bit of that, and a context that made the work we did (and it was surely a we) interesting to more than the usual suspects.
So how do you know? It's not always an off/on lightbulb. There is probably a range of time that qualifies as "ready to go". If you're doing an IDP (also see here and here), it should be very clear that they have checked off boxes that are (and should be) more than boxes to be checked. They have 2, 3, 6 papers (number being a function of project, years-post-college, and how much they knew when they started with you).
But one of the things that really makes me know that a postdoc in particular, but can also be a grad student, is ready to go, is when they are doing the mentoring on their own. They are working with the college students or the med students or the beginning grad students in ways that you know are Good and Right. The mentoring may not be exactly what you would do, or what I would do. They are different people. It is a Good Sign when a trainee does something different from you. Neither you nor I want to turn out Mini-mes, the world certainly has enough of me's in it. But when your trainees have figured out how to do the job, whatever it is, science, mentoring, teaching, with their own skills and mindset, then you have succeeded. You want to teach the algorithm, the language, not the answer.
I am not just proud of my trainee's first authored papers. One of my current trainees, who is well ready to run their own lab, is last author on a fairly important paper (recently totally accepted, and coming out in e-version Very Soon). Last authored because not only did said trainee do the work the science to get the paper out, the trainee got the first-author-trainee to produce a beautiful paper. I am buried somewhere in the middle of the author list. And damn, it feels better than the last authorship would.