Ideally you would run a separate conduit for ethernet cables across rooms since power cables can interfere if they are close. Running external is fine if you are okay with it (i.e. family). You can conceal like external wiring.
There are separate wall plates available for this, you can plug in keystones for ethernet. Apart from these some brands like legrand do have separate models for them as well. This is for internal wiring though. Will look odd for external.
