What is the real risk of salmonella with modern food cleaning standards?
If I eat 1,000 raw chicken breasts, how many terrible nights will I have?


Impossible to say. You will ingest many pathogenic bacteria, but nobody in the world can predict how many times they will cause an illness, and in how many of these times the symptoms will be noticeable, and how many nights each case will take until you heal. 

What is easy to measure is the actual contamination rate of meat. Here is one study which was done in American food: http://www.consumerreports.org/cro/magazine/2014/02/the-high-cost-of-cheap-chicken/index.htm. They found that 13.6% of the chicken breasts were contaminated with salmonella (out of a sample of 300). So, if salmonella is the only thing you are worried about, expect to come in contact with pathogens 130 times out of 1000 raw chicken meals. Again, the "falling ill" rate in these cases won't be 100%. 

On the other hand, being worried about salmonella exclusively is not wise. There are a ton of other pathogens on chicken. The same study found that 


  Almost none of the brands was free of bacteria


and, even worse, 


  About half of our samples (49.7 percent) tested positive for at least
  one multidrug-­resistant bacterium.


So, while you will be exposed to salmonella from only 13.6% of your hypothetical raw chicken meals, you will be ingesting pathogens with each of them. You won't fall ill every time you do it, but it's highly unlikely that it will be zero times either. 

This is just one article, but I have in the past read other studies which found similar numbers of contamination in other first world nations. 

I also recommend reading the whole article I linked, it contains much more interesting information than simply numbers on contamination. 



A side note on why I called your belief dangerous: We have very many food safety questions here, and I see all the time answers like "I ate it and nothing happened, so this practice must be safe". NO! Once you realize that unsafe food won't make you ill 100% of the time, and not even 10% of the time, you know not to mistakenly declare unsafe practices "safe" just because you got lucky a few times.