Let me preface this by saying that I don't have GPSmap units, I have the Echomap series. I looked at the Garmin website, the 8610 is listed as having 2 network ports (same as the Echomap Ultra units I have) you should only need to add a network cable to be able to view Livescope on the 8610 and the larger screen you add. You would network the displays by connecting them with an ethernet cable.
Keep in mind that your 8610 can only be networked with other GPSmaps units, not with the Echomap series.
Where people run into problems is with the 9" echomaps that only have 1 network port. They will sometimes buy multiple 9 in units, and LS only to find out that to network everything together they need a network switch to be able to connect more than 1 thing.
With 2 network ports you can daisy chain multiple devices. My set up is LS to 10" Echomap Ultra to 12" Echomap Ultra to 9" Echomap, I can see Livescope on all of the screens, share transducer and maps on all 3 displays. The 9" is at the end of the line due to only one network port.