Where do alpha testers download Google Play Android apps?

You need to publish the app before it becomes available for testing.

if you publish the app and the apk is only in "alpha testing" section then it is NOT available to general public, only for activated testers in the alpha section.

EDIT:One additional note: "normal" users will not find your app on Google Play, but also the activated tester can not find the application by using the search box.

Only the direct link to the application package will work. (only for the activated testers).

Here is a check list for you:

1) Is your app published? (Production APK is not required for publishing)

2) Did your alpha/beta testers "Accept invitation" to Google+ community or Google group?

3) Are your alpha/beta testers logged in their Google+ account?

4) Are your alpha/beta testers using your link from Google Play developer console? It has format like this: https://play.google.com/apps/testing/com.yourdomain.package

1. Publish your alpha apk by pressing the submit button.

2. Wait until it's published.
   (e.g.: CURRENT APK published on Apr 28, 2015, 2:20:13AM)

3. Select Alpha testers - click Manage list of testers.

4. Share the link with your testers (by email).
   (e.g.: https://play.google.com/apps/testing/uk.co.xxxxx.xxxxx)