It does happen and is a telltale that the assistant either doesn't know the proper technique for cutting sutures or was careless.
What you need to do is cleanse a pair of eyebrow tweezers and the finest, sharp pointed scissors you have, the ones out of a manicure set would be perfect. Or even a cuticle or small nail pliers - anything with a fine sharp point. Then cleanse them thoroughly just in hot soapy water - washing up liquid is fine - and dry them with a freshly laundered tea towel or hanky. This makes them about as clean as you can achieve.
Now wash your hands thoroughly.
Then wash the wound carefully with some freshly boiled water, cooled by leaving to stand in a covered bowl, with a drop of soap in it. If you have some Savlon, TCP or other antiseptic stuff would be good.
Firmly grasp the end of the stitch with the tweezers and pull on it as hard as you can bear. Then, with the sharp point of the scissors, snip the thread as low down as you can manage, even if it means going into the wound by no more than a millimetre. It might bleed a tiny bit but don't worry. Just cover it with a clean, dry dressing and press gently but firmly until it stops which should take about 4 or 5 minutes. Don't keep lifting it off to look as this means the clotting process has to start all over again.
Pulling on the thread means that when you have cut it, it should drop back under the skin level so it won't be felt. But this is why you need to be scrupulously clean about it.