You’ll want to continue to handle the paddle moving on the server because thats the most secure place to have it occur. You don’t want each client moving the paddle, even if the info comes from the server, because that could become vulnerable. What I would suggest doing instead if still having the paddle actually do the moving on the server, but also moving it on the client whose paddle it is. The delay wouldn’t even be noticeable by the opponent, and if the ball is slightly delayed in being hit, that’s also not a huge deal, it’s better to have that than to have an exploitable game.
I know this is not related but I was just wondering what do you use to move your ball?