There is a general rule, "Don't clean coins". Green stuff appearing on coppers, however, is an exception to thus rule.
First, we should point out that there are three different types of
"greenness" which might be visible on copper coins. "Patina" is what will gradually form on copper, over centuries. Patina is good, but your pennies aren't centuries old yet, so we can dismiss this one.
How are they stored? If you're storing the coins in a plastic album, it may be chemicals from the album itself which are damaging the coins. Believe it or not, some coin albums are actually dangerous to store coins in! If this is the case, the recommended treatment is a rinse with acetone or some such similar solvent.
FInally, there's the "green death" of verdigris - every coin collector's nightmare. If you're storing the coins "loose" in a jar or bucket, this is the likely culprit. I've got a thread going (well, OK, it's more like a blog-thing) on my trials and tribulations with getting rid of the "green death" from a badly affected copper token. Here it is.
I haven't had much luck with it so far, so for now, "do as I say, not as I do" may be good advice!
As to your sneaking supicion that the green gunk is contagious, you're quite correct - it can slowly spread across the coin's surface, and it can jump from coin to coin if they're stored together in a jar or container.
Your best bet for now is separating out all the ones showing even a trace of Green. For common-date, "typical" condition Aussie pennies where the green is winning the war, your best bet may well be to get rid of them; or use them to "experiment" with like I'm doing, so if you notice a more valuable specimen with the affliction, you can treat it.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis