I have only done this with 1 paid app, Read it Later Pro and it works. This method will work for pretty much all free apps.
1. Download the file manager app ASTRO, on your phone.
2. using the ASTRO app, backup the application. this will create an .apk of your app.
3. email the app, or use dropbox, or copy over the .apk to your kindle fire.
4. press and install the .apk. it should work.
if the app needs access to features that the kindle does not have, GPS, bluetooth, microphone, etc, there is possibility the app install might not work.
Also, i believe some paid apps that use the android market for legal authentication might not work. Also some apples that use you google account for authentication might not work (such as all the google default apps)