I don't think you need both, but I think you'd benefit from having both. I use LR for asset management: keeping track of my photos, tagging and sorting them, adding IPTC header data to them, and also for doing most of the tone and colour editing; and then when I want to do more than I can by dragging LR's sliders around and the Adjustment Brush won't cut it, I take the photo into Elements. I also like to use Topaz filters in Elements, even though I can call them up from LR; and the VirtualPhotographer filter which works in Elements but not it LR is great for adding effects.
For anything where I want to make changes to several parts of a photo but don't want to tediously brush everything with the Adjustment Brush, or if I want to make changes that the AB can't handle, I'll run the photo through Elements. When done I just press Ctrl+S to save and switch back to LR to find the edited .psd file there ready for more changes or for uploading, printing, etc.
It's a great and fast way to work, IMO. LR on its own is fast but not that flexible; Elements on its own is flexible but not that fast. And if you're seriously thinking about getting Elements, then do also check out Elements+ which adds back to Elements some of the best CS5 features that Adobe took out.