I split one of my 3 Joeys and will tell you how it works. Well, at least to me.
Basement. A big screen tv on one wall. Behind that wall I have an exercise room. Another tv is on the same wall. My Joey is underneath that smaller tv in the exercise room. That Joey provides HDMI and old fashioned RCA (yellow:video, White and Red: audio) ports. I use an HDMI splitter. So, hdmi cable coming from joey to splitter. 2 hdmi cables from the splitter are attached to both tvs. (I drilled a hole big enough on that wall for all the necessary cables.) Naturally, the hole is hidden behind big tv. When I watch the big tv in the basement, I am able to use Dish remote although Joey is in the exercise room. If my wife wants to exercise on the treadmill (which is rare!!), she reaches the splitter and presses a button (1 or 2. ) And she turns the smaller tv and Joey. It works without any issues. But, I can not use Dish on my big tv at the same time. I can use dvd/bluray players. I cannot even watch the same satellite channel on that big tv when she is using Dish. When she is done using exercide room tv, I just switch it back (using the hdmi splitter) to big tv.
I setup a bluray player for exercise room. She likes to use that. In this case, I am able to use Dish on my side with big tv.
Bottom line you won't be able to view satellite channel on both tvs at the same time. You can use either tv (one at a time) with a splitter.
Now, if there is a solution that provides watching the same Dish channel on both tvs at the same time, I would like to know, too.