Do people generally wear padded shorts when on a Peloton?

Or do the benefits of padded shorts for outdoor riding not transfer to indoor riding?

Saddles and Padding thickness is a very personal Item.

I spend quite a bit of time riding on the trainer, and in my opinion good shorts are more important indoors than they are outdoors.
I don't seem to change position nearly as often indoors and certainly feel it more on longer trainer rides.

Good padding and a good cream. I use bag balm for long rides. A saddle that supports your sit bones. This is important As you are sitting in one position for a longer time than a ride out doors. Also get good padded gloves and I use a good padding Handlebar tape. Move your hands around the bars during your ride. Overtime as you get adjusted you may want to invest in a rocker plate for your trainer.

Padding yes. I use chamois cream indoors as well but rarely use it outdoors. The lack of changing of positions causes a lot more friction issues indoors.

Yes, proper cycling shorts do provide a very real benefit. Quesiton: what saddle are you using on the Peleton? Hopefully not some wide cushioned vinyl PoS?

I think the Peloton specifically may be a little different to most indoor trainers. If you are doing the Peloton classes, then they're generally all an hour or less, and involve a good bit of out of the saddle riding. I found padded shorts to be much less necessary for this use case than when training on a regular indoor trainer or smart bike with longer durations and less out of the saddle time in general. It's the extended time in the saddle without a mostly fixed position that can make a chamois pad helpful for comfort. There's certainly no reason you can't use padded shorts for the Peloton but in my experience it's not all that necessary for doing the classes.

I wear padded shorts for trips to the corner store. It is just so much more comfortable. This reminds me I need to go through my collection and toss some of them. They are all getting stretched out.

It has the original saddle which is a decent road bike saddle with some cushioning.

Vinyl, leather, or something else? If he saddle doesn't breath a little it will cause issues. either way, a good pair of bike shorts will help.

It's a very personal choice, but I've had good luck with the Specialized RBX Pro shorts. Not a bad price, either.

Agree with you about the class length, but most Peloton classes are in the saddle almost the entire time. As such, I wear regular bike shorts for anything over 20 minutes. Compare that with SoulCycle, which have considerably more out of the saddle time. When I used to do SoulCycle, I'd wear regular workout shorts for hour-long classes with no problem. With the Peloton, I find I need the padding.

The benefits are exactly the same. I do a lot of indoor training and always wear my regular cycling bib-shorts. If anything I find the indoor trainer more demanding of my shorts (generally way more sweaty, largely fixed position with constant pedalling)

At least outdoors your are constantly shifting position. Indoors you barely move your butt.
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ditto on all points. Makes a big difference.