The way to view private photobucket albums is called "Fusking".
There are programs that help you view private photobucket albums really quickly, but you can also use the same method to manually view private photobucket albums and images too, one picture at a time.
Here's how it works...
How to view private albums on Photobucket
When a user sends a private image to their friends or shares it in private, the image location is sent via a url.
Usually it will look something like this: http://photobucket.com/image/user/DSC_002.jpg
The 002 at the end of the url is part of the file name. The next photo uploaded to photobucket might get the number 003.
But if the user only wants you to see 002, they will only send you that url.
But what is to stop you from trying 003 to see fi it loads? Nothing!
The problem is, even if the images are taken by the same user, and are from the same album, they still might not be sequential.
That's where a program comes in handy.
It can test out from 002 to 002222222 in a few seconds, finding all the possible images that user has uploaded privately.