There's certainly a mild risk to vaccinating for those three diseases, but the risk is much greater for not vaccinating. Especially once the population drops below the 95% vaccinated rate you need for herd immunity to work. For measles, for example, hundreds of thousands of children still die every year worldwide. (Interesting piece on vaccination resulting in a massive drop in measles deaths worldwide.
Yes, in the U.S., your kids probably won't die if they catch it, because they'll have access to better healthcare than the kids in third world countries: "The fatality rate from measles for otherwise healthy people in developed countries is 3 deaths per thousand cases. In underdeveloped nations with high rates of malnutrition and poor healthcare, fatality rates have been as high as 28%. In immunocompromised patients, the fatality rate is approximately 30 percent." (Wikipedia)
So let's say measles goes through your kid's partly unvaccinated class in the U.S. Thirty kids catch it, odds are, none of them will have a serious negative result. That's the kind of thing we grew up with. But if it sweeps through an entire school of a thousand unvaccinated kids, even here in Chicago, three kids will die. And some more will be left with brain damage from encephalitis, and blindness from corneal scarring.
Poor kids or those in rural areas in the U.S. might not have as good medical access, and even in the best case scenarios, there's still a higher percentage of deaths and other serious negative results from actual measles than the risks from vaccination. So for your kids, it's safer on average to vaccinate, even leaving aside the social consequences to others. That's why they recommend it, even though there's a slight chance that your kid might have a negative reaction from the vaccine. Most negative reactions are very mild. (I think so far, Kavi has had a mild chicken pox reaction to her vaccination; a bit of a fever and some red bumps.)
"Fever is the most common side effect, occurring in 5%-15% of vaccine recipients. About 5% of persons develop a mild rash. When they occur, fever and rash appear 7-10 days after vaccination. About 25% of adult women receiving MMR vaccine develop temporary joint pain, although this symptom is related to the rubella component of the combined vaccine. Joint pain only occurs in women who are not immune to rubella at the time of vaccination. MMR vaccine may cause thrombocytopenia (low platelet count) at the rate of about 1 case per 30,000-40,000 vaccinated people. Cases are almost always temporary and benign. More severe reactions, including allergic reactions, are rare. About one person per million develops inflammation of the brain, which is probably caused by the measles vaccine virus." (http://www.vaccineinformation.org/measles/qandavax.asp)And of course, when someone's child does develop inflammation of the brain, that's the case you hear about, over and over, on the internet. I can't blame those parents for their anger and grief. It's very hard to be cold about statistical risk, when you have direct experience of being the one who drew the worst possible hand.
It's tough for parents to make that choice -- if the risk of a serious negative reaction is 1 in 1,000,000, but you know you're risking that when you vaccinate, as opposed to the 3 in 1000 risk of measles death that you'll probably never have to deal with since you live in an area that's close to 95% vaccinated -- well, I can see why individual parents might decide to play the odds and let other families bear the risk. But the more parents who decide that, the greater the risk grows, for everyone. Very very quickly. "After vaccination rates dropped in northern Nigeria in the early 2000s due to religious and political objections, the number of cases rose significantly, and hundreds of children died." (Wikipedia)
The risk grows especially quickly for vulnerable populations. And you never know -- your kids might be in that group, they might have underlying weakened immune systems that you just don't know about yet. That happens a lot. Those especially at risk are those are those who can't vaccinate: pregnant women and their fetuses, infants, the elderly, and those with compromised immune systems -- kids who already have cancer, or heart problems, or other systemic issues that make it impossible to vaccinate.
Your family's willingness to take on the very slight risk that comes with vaccination, is what saves their lives, every day.