NO, you can have private and public albums.

Post the pic's you want to show on the forum into the public album and then you can put them in a thread.

You will have Photobucket (PB) open and right click on the image, select "Copy image location" and then in your post use the insert image icon (it will be the 4th from the left) and paste the url address. Once you figure it out, its pretty easy.
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If you think you can, you can. If you think you can't, you can't. Either way, you're right.