My own experience, and my sense from talking to other organizations, is that management time is a significant opportunity cost, while the benefit (the work done by the intern) is both highly variable in quality and potentially not that great in expectation - not just because interns may be less experienced, but also because you probably put in fewer resources into the selection process, commensurate to the expected duration, cost and amount of work the intern puts in (all low relative to a full hire).
Some organizations leverage internships better, typically in roles that require time but not a lot of experience/expertise, and that require minimal supervision - but precisely because of that, the internship because less valuable from a talent pipeline perspective (since by definition you already have an existing source of free labour).
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I work at 1Day Sooner, a small global health/biosecurity org. I'm answering based on my own observations, not in an official capacity.
We've had a couple of internships before as one-offs, but no continuous internship program. The biggest issues with running a standing internship program would be:
Speaking for myself, I actually would like to have more interns. I think good internships can be a great way to "pay it forward" in terms of all the educational experiences I've had that have gotten me here. I also just enjoy the opportunity to be a mentor! However, I mostly can't, in good conscience, argue that creating and running internships is a great use of my time or our grant money, and I don't think that we have enough internship-suitable projects that would make for a valuable and interesting internship experience. (I have offered to supervise/advise on independent projects before, which would be on my own time and wouldn't take up 1Day resources, although have yet to be taken up on that.)
Thanks for the helpful answer!
Can you elaborate on why it's not realistic for small organizations? It seems like a small organization could use an internship as a lead-in for a full time role, too. You can get a clearer signal about whether someone is a good fit for a full time job without committing to hire them upfront.
Sure, we're just not hiring for the same role repeatedly or on a predictable schedule, especially not entry-level roles. Like other orgs, we do work tests and trial periods, which are the fit-testing part of the hiring procedure. We often hire for pretty specialized roles, usually not entry-level, and right now we don't necessarily know the next role we're going to hire for, so we don't know what we'd want to look for in an intern. Those trial periods are useful lead-ins for the specialized jobs we've recently filled, they're just not what I would call an internship. When I think of internships-as-work-training, I think of the model at consultant, financial, or engineering firms, which know they're going to bring in new staff each year to fill entry-level roles that are very similar to each other, because they have a steady stream of work that fits a pattern into which they can predictably slot new people over the course of a year.
As David mentions below, 1Day also does periodically employ students or very early-career people for specific projects, which can look pretty similar to a paid internship in practice. But we can't know that we'll have a suitable full-time role available by the time those contracts are up. Basically, we sometimes do things that look like internships (and sometimes we even call them internships) and we also have a trial period for jobs, but there's not good overlap between what we have interns/student contractors do and what we've hired for lately.
Makes sense!
Worth noting that a number of 1DaySooner research projects I worked on or ran have paid some undergraduates, grad students, and medical school students for supervised work on research projects, which is effectively very similar to a paid internship - but as you mentioned, it's very hard to do so outside a well scoped project.