To be honest, I don't think most parents even realize that it was possible.
It is something that I was concerned about and looked into since DS2 was only about a month old when I had DS1 caught up on vax's. Personally I was reassured that while shedding from the MMR was theoretically possible, there were no acutal doccumented cases of a child catching measles from another child recently vaccinated with the MMR, and the only instances of Ruebella transmission I could find was a nursing mother being vaccinated and passing it through breastmilk, and I wasn't the one being vaccinate this time. Mumps I found nothign on, but Mumps certainly isn't common while unvaxed babies coming into contact with recently vaxed kids certainly is, so I judged the risk to be very low.
Chicken pox I was concerned about since I know that one can shed, so I talked to the nurse about it, and she said that that was why she'd already asked if I'd had chicken pox since DS2 was still young enough that my immunity would protect him. If I hadn't had my titers checked during pregnancy, I would have passed on that one, but since I knew they were high enough to consider me still immune I went ahead with it, and DS1 didn't end up with any pox or rashes or anything of that sort anyway.
I'm pretty sure they don't use flumist (or whatever the nasal one is called) here, or I would be concerned about that.
So basically I found the answers to my questions about shedding to be satisfactory enough that I wasn't concerned. But like I said, I don't think most people even realize that it is theoretically possible or actually happens with some vax's.