From what I've read, by they last quarter of the 16th century/first quarter of the 17th century the role of targeteers/rodoleros in European armies had significantly diminished. I'm wondering if they would have added pistols to their equipment by this time as a way to increase their tactical flexibility by giving them a short ranged advantage. I understand that in the early American colonies, pistol-carrying targeteers were not uncommon. Was the same true in Europe?

I thought the pistol was already rather common earlier on, even in Europe? Targeteers/rodeleros back in Spain often went around carrying a couple of darts/javelins, and as pistols became more common and reliable I think it wouldn't be out of the question to expect the richest sort of targeteers (officers, maybe? Monluc certainly carried the sword and target when leading an assault team into a town in mid-16th century Italy) to replace their darts with a brace of pistols.

In Elizabethen hands. No, not officially. Targeteers were part of the company orbat amd seem to have either led assaults, with the Captain and Lieutenant or acted as guards for the ensign. Officially the pistol was the preserve of the Horse, both Light Horseman and maybe some Demi lancers, not that Demi lancers were very common. It's possible that some Captains used their own pistols although the chivalry cult in vogue makes that slightly unlikely. It being somewhat unknightly to shot your opponent dead rather than beating him in hand to hand combat.. The pistol was also somewhat temperamental at this time, one senior Elizabethen Officer managing to shot himself in the face whilst using the barel of his pistol to hold open the visor on his helmet..........