Are there statistics about how many assertive people there are in the world? Based on my own experience, assertiveness is a rather rare commodity to be found among people, which in my opinion causes most of the troubles this world has.

Are there any statistics that attempt to measure how many assertive people there are in the world (or at least in some societies) compared to passive, aggressive and passive-aggressive people?
One method of estimating assertiveness would be to look at scores on self-report tests that aim to measure assertiveness. Measuring in this way suggests that assertiveness, as with other traits, is on a continuum. Thus, classifying someone as assertive or not assertive would involve a judgement call.
Nonetheless, once you have an estimate of the proportion passing such a threshold, you could then multiply population size by proportion classified as assertive to get a population estimate of assertiveness.

The NEO-IPIP has an assertiveness scale. But you'd have to hunt down some reasonable test norms. Alternatively, the NEO-PI would also be useful and the test manual would provide means and standard deviations for the corresponding assertiveness scale.

E3: ASSERTIVENESS (.84)
+ keyed 
    Take charge.
    Try to lead others.
    Can talk others into doing things.
    Seek to influence others.
    Take control of things.

– keyed 
    Wait for others to lead the way.
    Keep in the background.
    Have little to say.
    Don't like to draw attention to myself.
    Hold back my opinions.


Herzberger et al (1984) provides one simple example. 
They created what they called the Assertiveness Self-Report Inventory. 
It included items like


  
  When my date has acted rudely at a party, I don' hesitate to let him/her know I don'd like it. 
  After eating an excellent meal at a restaurant, I do not hesitate to compliment the chef.
  When people use my car and don't refill the tank, I let them know I feel unfairly treated.
  


Overall there were 25 items True/False items like this. Using a relatively small university sample (around 150 all up),  mean number of items endorsed was around 10 and the standard deviation was around 4. You could potentially look more carefully at such items perhaps using expert raters of scores to generate a threshold above which constitutes an assertive personality. For sake of example, you might determine that participants who have endorsed  12 items or more are deemed to be "assertive". If you had the raw data, you could determine what sample endorsed 12 or more items. Failing that, you could get an approximate estimate using a normal distribution approximation based on the mean and standard deviation or some variant on this approach.

Of course, there are many limitations to this approach:


Self-reported personality is not the same as behaviour or some more objective or other report measure of personality. 
defining the cut-off can be somewhat arbitrary. 
You would need to define more clearly how you want to conceptualise assertiveness, in that you seem to be distinguishing a more adaptive form of assertiveness in contrast to a more aggressive approach. 
Getting representative and large norms for even one country is difficult, let alone the entire world. That said, the NEO-PI and IPIP are likely to offer reasonable existing sources.
As others have mentioned, cultural norms should have a dramatic effect on what constitutes reasonable ways of asserting your interests in social ettings. This sociological perspective may even provide a better understanding the phenomena of interest.


References


Herzberger, S. D., Chan, E., &amp; Katz, J. (1984). The development of an assertiveness self-report inventory. Journal of Personality Assessment, 48(3), 317-323.