Is there a way to set up Android's "Slide" lock screen in such a way that swiping to different directions changes the background image to a specific image? Such as, swipe left for a personal image, swipe right for a professional image. 3rd-party apps would be acceptable as well.
I've never heard of this. However, you may be able to set it up if you have a couple of different home screen launchers installed.
Some lockscreens let you open a specific app depending on where you drag the unlock icon. You could set it up to open one of your launchers depending on which side it gets dragged to.
Then, just set the background for each launcher. Note that, with separate launchers, you'll have separate home screen layouts as well. If you want the same apps on the home screen, you'll have to manually duplicate your effort of setting up the home screen on both. Of course, this may be an added feature for you, as you could have business apps on one and person apps on the other.