Plenty of such devices. Most are fairly expensive.
If you look at the diagrams for the devices linked you see that there is still a source, the sat box in this case, feeding all of the monitors that are linked together by the RG-6 cables. What this does is allow you to send the same picture you are watching in the family room to monitors in other parts of the house, probably over the cable in the walls of your house. However, in this configuration, everyone is limited to watching the same channel. That's fine if that is what you want.
That is not going to work on the input to a sat box, if this is what you're contemplating. There are DC currents in the line between the dish and the transceivers in the sat box, at different voltage levels on DTv boxes, that are used to select which channel is viewed. Also, the receiving end must have a transceiver to "talk" with the satellite, and decoders to handle the format that the content is broadcast is transmitted in. I haven't kept up with Dish, but DTv is mpeg 4, IIRC. Your monitor, or tv, would still need to have a way to work with the dish.
Thank you for the responses. To clarify, my Dishnetwork receiver has two tuners with two remote (IR and UHF). I have hooked the receiver with an HDMI connection to a HDTV and the picture is fabulous. However, a secondary output is coax only and I have it sent to a spare room. I had a regular tv there to which the coax output from the tuner was hooked up. Using the UHF remote we could watch the Dishnetwork receiver in the spare bedroom. Well, my son took the tv to college and ended up with another monitor which has only HDMI input. Somehow, I need and inexpensive way to hook up the output from the Dishnetwork receiver into HDMI so that I can connect it to the TV. Hope all of this makes sense. Appreciate any advice.