You just do LiveView instead of a regular controller. No duplication.
When you request a page, it is render on the server and all of the HTML is returned over HTTP as usual.
After the client has received the HTML updates, live updates can go over a websocket. For instance you start typing in a search field, this is sent to the server over websockets. Then the server might have a template for that page that adds search suggestions in a list under that search field. The server basically automatically figures out how the page should be rendered on the server side with the suggestions showing. By "re-rendering" the template with changed data used with the server side template. Then it sends back a diff to the client over websockets. The diff adds/changes the search suggestions to the page. The diff is very small and it's all very fast.