Seems mostly the latter, but that's true even for the companies that hire citizens too. My dad works as a drywall foreman, and they'll end up hiring a mix - some experienced guys, some guys with absolutely no training (they don't hire illegals though). The guys with no experience start only at minimum wage. Some of the more experienced guys will tell them what to do at first. Typically they start out on the simpler tasks - on drywall this being things like "hanging" (affixing to the wall) or especially sanding. Finishing (puttying the screw holes and seams) is a bit tougher and so they don't start there until they've been doing they other stuff for a good while. If they express interest in finishing they'll sometimes let them start practicing a bit in an inconspicuous area - a closet or something where any mistakes they make won't be as visible.
If after 3 or 4 months the person seems to be catching on they'll keep them and let them start doing different things as time goes on. If they don't appear to have a knack for it then they'll let them go. Either way though it's a job that people seem to get into pretty often without any formal training - just on the job. I'd suspect that many illegals end up the same way. Start out simple, and if they work out then they stay. Whether here or there everybody has to start somewhere.