TLM's right about the spring.
If it is that kind of stud, have a look and see if the spring is actually still there. Sometimes they crack and the little pieces fall away unnoticed. If that has happened you can butcher a new one for it's spring.
It's not easy to fit the replacement, especially if it's a rivet type rather than a sewn on one, but it is possible.
I make jewellery, and the tiny little pliers are ideal, but my husband's electronic pliers are just a neat.
I have used decent tweezers before now to do the job though.
If it's not that kind of stud, if it's one of the KAM type, then you can sometimes get them to tighten up by gently distorting the inner protrusion. Make it a little more oval rather than round.
I like the sound of Robson Valley's tiny punch marks though.
If it's something worth the time, something you'll get a lot of wear or use out of, it's generally easier to just replace it.
The Shoe Repair (Timpson's for instance) folks have a machine that will fit one for you.