If my family has multiple iPhones that use the same Apple ID, is there any way to not share Apps?
From your iPhone go to Settings > Store and disable Automatic Downloads for Apps. You can elect to do the same for books or music.

Your sons can visit the App Store and download any app that you've already purchased without being charged, though they will have to know the login information for the Apple ID tied to your iTunes Store account. 

The alternative is to specify that the applications be synced to their phones from within iTunes.


  To sync downloaded apps to iPod touch, iPhone, or iPad: Connect your
  iPod touch, iPhone, or iPad to your computer.
  
  In iTunes, select your device (below Devices), and then click the Apps
  button.
  
  Select Sync Apps, and then select the apps you want to sync.