Screen mirroring and private listening (via the Roku app) should work fine on a network without Internet access, as long as the network is properly configured to use a private IP address range and there's no security setting in the access point that blocks connected devices from seeing each other.
The main issue is, without Internet access what do you plan on using your Roku for? If you have a home media server, such as Plex, Emby or Serviio, access to the media on those servers should work fine, although those servers will be unable to gather metadata from the Internet sources they use to provide that information. And for Roku devices with a USB port (your Premiere does not) you could connect an external drive with media and play that. But that's the limit to what a Roku can do without an Internet connection.
Also, be aware you cannot connect a Roku the first time to a wireless connection that does not have Internet access, as it will fail the connection. But if your wireless connection has the same SSID and password as a network that does have Internet access, then you can connect it to the Internet access point first, and the Roku will then connect to the access point without Internet, as it doesn't check for Internet access for subsequent connections to the same SSID.
Dan
Roku Community Streaming Expert
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