From what I understood the crossbowmen themselves were professionals who were paid a fair bit and brought armed retainers with them.
"Professional" yes, but not in the modern meaning.
Originally, the crossbowmen were sailors from Genoa that were able to make quick attacks on the coast (using ships as siege engines as well). In time of peace, unemployed they searched for someone wanting to pay their services.
So, already there, we don't have real "professional" origin. But as usual, guys that learned about weapons during war and not finding a job after, or simply not willing to find one.
Furthermore, they quickly escapade the republic's grasp, that (as said above) was unable to refrain companies to form and to be hired, even by ennemies of the city.
They seems to have been less plunderers and raiders than the routiers, it's true, but it's maybe due to their specialisation that assured them to find work quickly.
Would those be the "pietons"?
It can define any non-noble footmen that wasn't a sergent (that are standing soldiers tied to someone by feudal link), an archer or a crossbowman.
Not sure if it understood the workers (as masons) though. It's possible but I don't have a clue. Probably it was sometimes counted as, and sometimes not.
So usually, "piétons" were pikers or equivalent (wielding weapon issued from rural world, often agricultural tool transformed into war weapons, then forged as war weapons).