Not usually, and not really.
Firstly, acupuncture needles are completely different from hypodermic needles. In acupuncture, we’re not inserting anything foreign into the body other than the needle itself. Hypodermic needles are hollow so that something can be administered (immunizations, fluids, etc.) or withdrawn (blood.)
Acupuncture needles, by contrast, serve only to stimulate the point into which they are inserted. They don’t insert or remove anything, so they don’t need to be hollow. They are solid, very thin, and flexible.
Secondly, “pain” is more interesting here than in other situations. I promise.
Pain from an acupuncture needle is nothing like other kinds of pain you may think of and want to avoid– stubbing your toe, burning your hand, dry-heaving, or breaking a bone. If there is pain, it usually subsides either instantaneously or very quickly morphs into something else.
That something else is highly personal. People report different sensations regularly – heaviness, tingles, a kind of hum or buzz, warmth, coolness, a flowing feeling, or nothing at all.
In general, there are certain acupuncture points that get a reputation for being sensitive. The tips of the fingers, for example, are not very fleshy and stimulation by an acupuncture needle can be strong for some, but not all, people. Some barely notice – which may or may not be a sign of health. Needles in hips and abdomens often go unremarked-upon.
Furthermore, points are often sensitive when the person has a problem area associated with that point. For example, there’s a point on the hand that is associated with the back of the head and the spine. If I needle it on someone who has occipital neck tension, the point is often very strong and tender. If I’m needling it simply to access the spine and the person’s neck is not especially painful, the point usually goes unnoticed.