I dunno how it goes in the US or Canada but here in Hong Kong (I study in the Hong Kong Academy for Performing Arts), ASM do almost everything.
But it depends on what show you are in, like for my previous show, a dinner show celebrating a major local magazine's 40th Anniversary, ASM (aka Project Assistants) do a lot of paper work. During pre show we did researches on costumes (for a dance piece in the show) we did pricing for decoration accessories, we made powerpoint presentations of the magazine's past glories for the show night...and during the dinner show, we acted as escorts to make sure the guests go on stage and off stage safely, we prompted the dancers to go on stage, and I even learned to sew elastic straps on high heel shoes coz the dancers need to dance in them.
As for theatre shows, we mostly do props research, prepare rehearsal rooms, shop for stuff, help with scene changes..and loads
Well but coz every show is different, do more shows and get more field experience is the best way of learning.