Depends on the nature of the piercing blow. Let's say the riveted mail turns a thrust of a sharply pointed sword, wielded by a warrior. The target would no doubt feel the blow, but wouldn't be unduly injured (see the comment above about the gambeson, it's an essential part of the gear).
If the piercing attack is an arrow, I doubt it would do any injury if it failed to pierce. A lot depends on the force of the arrow, how powerful is the bow, is it a close range, direct hit, or a long range shot, or a glancing shot, etc. The target might not even notice. I recall a description of Crusader troops at the Battle of Arsuf, harassed by long-range arrow fire from Saladin's troops, some of the mailed sergeants had arrows bristling all over their mail, but were not seriously injured. You got a different result when Turkish mounted archers fired directly into charging Crusaders at close range (eg Hattin).
Now if the attack is a sharp lance, backed by a mounted warrior on a charging war horse, besides being unhorsed, possibly having a broken rib or other injuries, the target is probably thanking his deity he survived.
Mail is pretty much proof against cutting (it's still the most cost effective way to protect meat cutter's hands). It's pretty good against piercing (the rings have to be riveted). But it's no protection at all against crushing blows.
Early Medieval panoplies tended to center on sword/spear combinations. As mail becomes more widespread, you get more interest in crushing/chopping weapons, maces get popular about 1100. As soldiers add layers of plate (say a Visby-style coat of plates or full back & breast up to Gothic armors), the crushing weapons get more... crushing. A flanged mace does some mashing, compare that to a 15th century war hammer. One's for potatoes, the other is for cracking a lobster.