In the professional world the SM wouldn't be the person making those calls, since it's a hiring situation. In community theatres, though, there's a huge amount of variety. Back in my community theatre board member days, it was the responsibility of our directors to call their casts. Sometimes these shows didn't even have a stage manager yet (yes, really).
Here's a pro tip from those days: we would make phone calls beginning with the leads and working our way down, waiting to call the "not cast" list last. If for any reason someone bows out of the show at this point (which, especially in the community realm, is prone to happen), you don't have to un-un-cast someone, and can move people up without anyone being aware of a problem. We were especially mindful of this situation when working with children casts.
Speaking from personal experience: don't assume actors will check back the website if they don't hear from you personally. I was once cast in a show and didn't realize it until I checked the website a few weeks later to see if they were selling tickets yet, just a few days before rehearsals were set to start.