Thanks for the pointers; that gives me something to check next time. I'll watch the EICAS while starting (a bit awkward in the sim, but possible).
I've seen yellow notations on the EICAS before after engine start that disappear after a moment, but since it's hard to watch the overhead panel and an EICAS display at the same time, I don't know if they occurred simultaneously with the aural signal. I'll watch both henceforth just to see if something appears the next time I hear it.
From what I gleaned from the sim documentation and real-world docs, I autostart an engine with engine bleed ON, fore and aft fuel pumps for that engine ON, engine hydraulic pump ON, demand AUTO, generator ON, fuel cutoff ON, and finally engine start button pulled (some sources say I should pull the start button first and then cutoff, but maybe it doesn't matter for an autostart). I do this identically for each engine. Numbers 4 and 3 usually start silently, but sometimes I get the aural signal for 2 and 1.
I'll report whatever I find out, and then perhaps someone can tell me if the sim is behaving like the real thing. I'll try some manual starts, too, to see if that makes any difference.