Find the number of downloads for a particular app in apple appstore

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Outdated answer: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!

xyo.net

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.