One server with devices on two networks

I have a Plex server on a Linux box with two network cards. My home network is 10.1.1.x with a Roku on the network. The second network is 192.168.11.x with a Playstation on that network. Both the 10 and 192 networks have cable modems attached. In addition to myself the Plex media server has two managed users. I would like a user on the 192 network using the Playstation to stream media over the 192 network. However the Playstation is being served over my external network so data is being uploaded from my cable modem and pulled down from the cable modem on the 192 network.

How do I get the Playstation to connect to the Plex media server using its local network? When it brings up the Plex app it wants an authorization code which I generate. It does not seem to find the Plex media server on its network. Here is a diagram of the configuration:
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This is what netstat shows when the Playstation is streaming media:

tcp     0      0 0.0.0.0:32400          0.0.0.0:*               LISTEN    885/Plex Media Serv 
tcp     0      0 127.0.0.1:32401        0.0.0.0:*               LISTEN    885/Plex Media Serv 
tcp     0      0 127.0.0.1:58459        0.0.0.0:*               LISTEN    1198/Plex Plug-in [ 
tcp     0      0 10.1.1.241:58448       45.79.85.134:443      ESTABLISHED 885/Plex Media Serv 
tcp     0      0 10.1.1.241:32400       47.33.162.170:60712   ESTABLISHED 885/Plex Media Serv 
tcp     0  86880 10.1.1.241:32400       47.33.162.170:57500   ESTABLISHED 885/Plex Media Serv 
tcp     0      0 10.1.1.241:32400       10.1.1.100:33940      ESTABLISHED 885/Plex Media Serv 
tcp     0      0 10.1.1.241:32400       10.1.1.100:33926      ESTABLISHED 885/Plex Media Serv 
udp     0      0 0.0.0.0:32410          0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 0.0.0.0:32412          0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 0.0.0.0:32413          0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 0.0.0.0:32414          0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 10.1.1.241:44825       0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 192.168.11.244:53100   0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 127.0.0.1:37021        0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 192.168.11.244:58708   0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 0.0.0.0:1901           0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 192.168.11.244:59818   0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 10.1.1.241:47560       0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 10.1.1.241:39840       0.0.0.0:*                         885/Plex Media Serv 
udp     0      0 127.0.0.1:36179        0.0.0.0:*                         885/Plex Media Serv 
unix  3      [ ]         STREAM     CONNECTED     18752    885/Plex Media Serv

This shows that the Playstation on the 192.168.11 network is using the cable modem and streaming data from the 10.1.1.241 network adapter. If I start streaming with my roku a netstat will show the following additional entries:

tcp    0  71904 10.1.1.241:32400      10.1.1.111:45336      ESTABLISHED 885/Plex Media Serv 
tcp    0  48672 10.1.1.241:32400      10.1.1.111:45345      ESTABLISHED 885/Plex Media Serv
tcp    0      0 10.1.1.241:32400      10.1.1.111:53538      ESTABLISHED 885/Plex Media Serv

In that case the roku is also pulling data from the 10.1.1.241 interface and keeping traffic local. The computer that is the Plex media server is able to connect to the cable modem attached to the 192.168.11.x network as I have been able to browse the cable modem status page (using an ssh tunnel to get the browser on that network).

I would like to get a better understanding of Plex networking and figure out how to get the Playstation to find the Plex server on the 192.168.11.x network.