Normally the right engine-only (left engine failed) case requires more rudder input to correct for asymmetric thrust than the left engine-only because propeller P factor results in more rudder effectiveness when flying on the left engine (the net thrust line of the left engine, being offset to the right of the propeller disc, is closer to the rudder so the rudder has less work to do, and farther from the rudder on the right side so it has more work to do).
Unless the engine has counter-rotating propellers, in which case P factor effects are symmetrical, so the rudder requirements are the same in an engine-out situation.
Beyond that, in a twin with wing mounted engines generally, to achieve coordinated flight you bank slightly into the live engine (normally 5 degrees bank into the live engine is rule of thumb used for multi-engine training).
The reason you bank is that the thrust from the live engine, summed with the lateral thrust from the rudder application, causes a resultant thrust line that is skewed toward the dead engine, and you fly slightly sideways with the wings level. Banking into the live engine introduces a sideslip component in the opposite direction, toward the live engine, realigning the resultant thrust line back to (more or less) straight ahead.
The result is the airplane is flying slightly banked, but straight through the air, that is, no sideslip even though the live engine wing side is down a bit. The skid ball will be pointing to the center of the earth but that location of the ball in the vial is toward the low wing (the ball should be about half a ball width off center toward the live engine).
Basically, the sideslip is cancelling out the lateral thrust caused by the rudder displacement, or conversely, the rudder input is countering the sideslip tendency from the slight bank being applied.
You can bank more than that, and it will require even less rudder, but you will now be side slipping toward the live engine and if you are in a light twin on a hot day with a lot of junk on board, you won't be climbing until you get things corrected.
So more rudder displacement is required, all things being equal, when the "critical" engine has failed, usually the left one if the engines both rotate the same direction, clockwise from behind. If the engines are counter-rotating, like on a Lockheed P-38, or the Piper twins that had that feature, there is no critical engine and the displacements and forces are the same for either engine, although you still should be banking slightly into the live engine to maintain true coordinated flight.
On jets with fuselage mounted engines, this banking toward the live engine isn't required and is not taught, I expect because the amount of bank required is only a degrees or two, not enough to bother with.