As for the latest gen of Slingboxes (350, 500 renamed Sling TV, and M1): yeah, they do make initial contact via internet. In fact, that is a big irritation to some Slingbox owners on the Sling forums because even if they just want to stream via their LAN, if you do not have internet, you've got no Sling. So, if you lose internet for a period like an outage, you can't Sling on your LAN to entertain yourself.
Also, if you choose to view your Sling content on a PC or laptop you must use a browser (the older PC apps for viewing are NOT supported by the new generation), and for both PC/laptop and mobile device, you must sign on to your Sling account first: you need internet. In the case of a smart phone, your can use your cell phone service provider's network to log on to your Sling account and then commence Sling viewing on your cell phone network. Of course, this will really eat into your data usage for the smart phone.
However, there may be a Slingbox solution in the case of viewing Sling content on your Roku. The Roku Sling app is limited to functioning with smart phones ONLY, for now.
First add the Sling app from the Roku channels store. The app is FREE (unlike for the Sling apps for other devices). I think the Sling app may only function or is available on the later generations of Roku, so please research that point first. You begin the process by using your smart phone to access your Sling account and content via your smart phone (using the cell phone network). Then once your content is Slinging to your smart phone, you can select to "cast" the content to your Roku. That will cause the Slingbox to drop the connection to the smart phone and, instead, send content directly FROM the Slingbox itself to the Roku, wherever it may be. If the Roku is on your LAN, then you are now Slinging on your LAN to the Roku. At this point your smart phone now becomes your Sling remote control, and I believe is the only means to control your Sling experience on the Roku. However, you are no longer using your cell phone network to Sling your content and nor are you any longer using your allowance of data on your smart phone plan.
Try this out and see if it works for you. Let us know what your finally do.