Same dies work fine
I do not find any need for different die sets when loading sister calibers such as .38/.357 - the .38 dies work perfectly for both.
If you are asking about different BRAND dies instead of two sets - no, I absolutely recommend Lee dies for pistol calibers. Other brand dies work fine too, but only the Lee gives me the easy capability to use their fine through-the-die auto powder measures that is a REAL benefit when loading large numbers of rounds. You do not have to use their through-the-die powder feature, but it is always an option that you do not have at all with other types of dies. (I do not use that feature at all on a single stage press, but on a progressive or turret press, I would not live without it.)
Where a second die set gives you value is in not having to change the adjustment settings if you use a turret system or when you switch between calibers in a single stage press. With a different set of dies for each caliber, you can adjust them one time and then leave them all locked in. It is a relatively minor issue, but if you switch back and forth a lot, the convenience is nice.
By the way, if you are only going to be using a single stage press without a quick-change bushing (where you have to always screw the full die body in and out between each stage), you will probably want to buy a set of locking rings to replace those o-ring locks on the Lee dies. A lot of people find the standard Lee o-ring locks easily change adjustment when spinning in and out.
I personally love my Lee dies with the o-ring system in my turret presses, but for single stage use I would always use a split locking ring for absolute security of holding the adjustment. Do not buy the Lee Lock Ring eliminator unless you have a Lee press that specifically needs that system. I prefer the split lock rings from either Lyman or Hornady. RCBS lock rings are just fair in my opinion, since the lock is only a set screw instead of a split ring that clamps on the entire body.
I have never used the Hornady quick change bushing system, but it looks real nice to me. If I was setting up a new set of Lee dies to use in a single stage press, I think I would buy their "Hornady 44099 Lock N Load Conversion Kit" instead of new lock rings - with those bushings, the Lee o-ring lock will be just dandy.