There is no way to know unless the particular company reveals the info. The best you can do is find a few companies that are sharing and then extrapolate based on app ranking (which is available publicly). The best you'll get is a ball park estimate.
Updated answer now that xyo.net has been bought and shut down.
appannie.com and similarweb.com are the best options now. Thanks @rinogo for the original suggestion!
Outdated answer:
Site is still buggy, but this is by far the best that I've found. Not sure if it's accurate, but at least they give you numbers that you can guess off of! They have numbers for Android, iOS (iPhone and iPad) and even Windows!
I think developers can do this for their own apps via iTunes Connect but this doesn't help you if you are looking for stats on other peoples apps.
148Apps also have some aggregate AppStore metrics on their web site that could be useful to you but, again, doesn't really give a low-level breakdown of numbers.
You could also scrape some stats from the RSS feeds generated by the iTunes Store RSS Generator but, again, this just gets currently popular apps rather than actual download numbers.
found a paper at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1924044 that suggests a formula to calculate the downloads:
d_iPad=13,516*rank^(-0.903)
d_iPhone=52,958*rank^(-0.944)
Source: Stackoverflow.com