Is it possiblde to have both going at the same time running from the same network share?
Nothing to stop two or any number of WD TV Live from all of them accessing a Network share simultaneously except performance. The computer or NAS device serving the Network share may not be capable of handling high bandwidth service to many requests, and eventually the network bandwidth and router will choke if the traffic exceeds its capabilities.
There are bits moving around and only so many of them can any network handle. You need to understand where the limits are in your own situation… Al