It depends on the load.
FMJ poses no issues, and some, indoor ranges require bullets with no exposed lead (it depends on both the type of backstop and their ventilation system).
Lead round nose bullets can be a bit smokier due to the bullet lubricant burning. If the range doesn’t have pretty substantial ventilation at the firing point that smoke can be noticeable.
At the target end, if the backstop is a big pile of ground rubber they probably won’t care, provided they have sufficient ventilation at both ends. If they have a bullet trap (usually two large angled plates that skip the bullet into a gap between them, were the bullet then enters a circular trap), they might have concerns about exposed lead - or not. Again it depends on ventilation keeping lead particles to non detectable levels.
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Load wise, lead bullets in .38 Special generally do not create any significant leading issues in a properly functioning revolver.
In a .357 Magnum lead bullets can be a much bigger problem. The bullet has to pass through the end of the cylinder and the forcing cone before it engages the rifling and seals the gas behind it. With .357 Magnum pressures, you can get some gas cutting before the bullet obturates in the bore, and that gas cutting removes lead and potentially deposits it in the bore.
Another potential lead bullet .357 Mag problem is the higher velocity the bullet has when it reaches the rifling. If the alloy is too soft, it can skid a bit in the bore before the bullet starts to rotate in the rifling and that will lead the bore up in a major hurry.
It’s a bit of a balancing act. You can’t have an alloy that is too soft at .357 Mag velocities and pressures or you’ll get leading. However you also can’t have an alloy that is too hard as then it won’t obturate in the bore and you’ll get gas cutting and leading.
Consequently it’s common for cast bullets at .357 Mag performance levels to have a gas check to prevent gas cutting.
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I used to cast bullets a lot for several different calibers, partly for economy, partly because it was the only way to get top quality bullets for long range shooting with my Sharps, and partly because I enjoyed the challenges involved.
But now I just buy commercial cast bullets for a couple .38 special loads and use plated bullets for pretty much everything else where I used to use a cast bullet. It solves the leading issues and eliminates the smoky lube issues, and they don’t cost much more than cast bullets.