Ideally, you would 'home run' coax. For example, I have an 8-way splitter in a wiring closet, in my basement. From the wiring closet, I have run two coax cables to the upstairs living room, to F-type wall outlets. This gives me the flexibility to feed separate OTA-TV and OTA-FM signals to the TV and FM tuners. All of the coax, phone and data cables originate at the same wiring closet.
If you already have a single RG-6 coax run from the splitter to the location of the new LAN tuner, and it's impractical to run another coax, you can use a 2-way splitter adjacent to the new dual tuner. It will work just fine, though you will only have one signal source available to the two tuners and you will loose anther 4 dB of signal level through the 2-way split. For most applications, the end user is looking for the same signal to each tuner, so, no worries.
My use of the Silicondust HDHR tuners is somewhat unusual.