In-app purchases made in the normal versions of the Angry Birds games are not transferable to the HD versions and vice versa. Unlockable in-app purchases, such as the Mighty Eagle are transferable between iOS devices that have the exact same version of the game installed. The exceptions are all consumable in-app purchases, such as power-ups, which are not transferable.
When you download the Mighty Eagle or other unlockable item onto an additional iOS device, you will not be charged again. Click the shopping cart from the game to open the shop and scroll to the right. Once you see the blue Eagle eye surrounded with green arrows, click it and you will be asked if you wish to restore purchases from your iTunes account. Select yes and fill in your account info (if asked) and your unlockable item will be added back to the game. Of course you must be using the same iTunes Store account (Apple ID) that you originally used when purchasing the item to get it back.