Keep in mind that the "cloud" version of Lightroom does not store anything on the internet. All your images are locally on your hard drive; all the software is installed on your hard drive. The "cloud" version just lets you share your images via Adobe web services with the public or your mobile devices, if you choose to share them.
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Best to use the 30 day trial so you can understand the difference between Lightroom and Photoshop Elements. Also even for the stand alone version of LR 6 you require an internet connection to register and activate the product. One activated you will not have to maintain an internet connection as long as you do not Sign Out.
Even with Lightroom there are lots of features in PS Elements 9 that will come handy for advanced editing that are not available in Lightroom.