I'm very interested in this topic as I am weighing whether or not to get my 2 year old twins vax'd for measles right now.
My first reaction is: when he says "1 in a million vs. 1 in 1000" that is a faulty comparison. If you get the shot, you have a 1 in a million chance of getting a severe side effect (I know it is probably much higher, but for the duration of this argument I will accept his number). However, if you do not get the shot, you do NOT have a 1 in 1000 chance of getting brain damage or dying. That's because you may never get measles in the first place. So he needs to take into account what the chances are that you will even get measles, then he can apply his 1 in 1000 statistic.
(BTW, I can't find anything on hearing loss being caused by measles, as your friend stated. I saw hearing loss associated with MMR vaccine though. That might be because mumps and rubella can cause hearing loss?)
Now I am going to take you through my layperson’s projection of an unvax’d child’s chance of dying or sustaining serious injury from measles. I don’t know if I’m doing this right, so I’m going to be detailed and spell everything out. I’m also going to err on the conservative side – giving the pro-vax side a large benefit of the doubt. All my numbers are from the CDC pink book 10th edition.
First, some history. The last big outbreak in the US was in 1989-1991. There were 55,622 reported cases,* "mostly among the unvaccinated". Of those 56K reported cases, 123 people died. That's higher than your friend’s 1 in 1000 number -- it's 2.2 deaths per 1,000 reported cases. Roughly half (49%) of those who died were under 5 y.o. The biggest cause of death for children is
pneumonia, not acute encephalitis -- 60% of child deaths are from pneumonia. I thought that was really interesting because you never hear that anywhere. I guess acute encephalitis sounds much scarier than something as mundane as pneumonia.
* For reference, US population was 248,709,873 according to the 1990 census.
Now, my unscientific projection. Let’s say we have an outbreak the same size as the ’89-’91 one. We would have ~67,404 reported cases (22 per 100,000 applied to current population size of 306,382,859). If we apply the same death rate of 2.2 per 1000, that's 148 deaths. I don't know how many are brain damaged (I think I read that acute encephelitis has a 15% fatality rate and a 25% brain damage rate), but I think it's more than fair to just double that number and say approx. 300 deaths or permanent brain damage.
So
what is my unvax'd child's chance of dying or suffering permanent brain damage from measles, in an outbreak of this size?
We can’t just take 300 and divide it into the total population number, because that does not account for the fact that an unvax’d person is “35 times more likely” (according to your friend) to get measles than a vax’d person.
If I assume that 1% of the population (just over 3 million people) is not vaxed, and that all 300 (not just 93% of the 300) who die or sustain serious injury from measles are not vaxed, then I think the chance of an unvax'd person dying/injured from measles is 10 in 100,000 (300 divided by 3 million).
If I assume that 5% of the population (15.3 million people) is not vaxed, then the chance of an unvax'd person dying/injured from measles is 2 in 100,000 (300 divided by 15 million).
If I assume that 20% of the population (61.3 million people) is not vaxed, then it’s <0.5 in 100,000 (300 divided by 61 million).
I have no idea what the current measles vax rate is. In the ’89-’91 outbreak, CDC says in some of the outbreak areas, the vax rate was as low as 50%. You can see from the above that the higher the no vax rate, the better the statistic sounds for no vaxers because you can divide the death/injury rate into a larger pool of people.
In any case – as best as I can figure, if I accept the 1 in 1 million vaccine injury rate, then my child is better off being vax’d. Although I do not necessarily accept that vaccine injury rate, I am seriously considering getting my kids vax’d for measles because a 1 in 500 death rate (2.2 per 1000 reported cases) makes measles a pretty serious disease for me. I know the true rate may be lower as not all cases get reported. But still. Out of all the shots, this is the one I’m considering most seriously so far.
Well thanks for reading with me, if anyone made it to the end!
(EDITED to add

Thank you to Sirelee for the stats from other sources... 1 in 9,000 death rate sounds much better and that is from recent data (2001-2007 in Western Europe). I think I might have just flopped back to the no-measles-vax side for my kids...