can't do auto-starts when the timer is on another pc, for that you have to have the timer running on the same computer. if you're doing manual timing (a.k.a. hit split/start/stop manually) you can just use some input device to send those signals to the pc the timer is running on.

I know this is a bit old, but I've just started creating a LiveSplit plugin that might help this situation. It uses the LiveSplit server function to sync two instances of livesplit. It's ideal if you use 2 PCs to stream but still want to use an in game timer or autosplitter

Here's a link to the github page. The readme has all the instructions https://github.com/Villhellm/LiveSplitClientPlugin