I've had peppers cross. Not such a biggie when it's something like two colors of a similar swet pepper or heat level of hot pepper. I've seen some color mutations or shape irregularities. Not so great if I want a pepper for particular qualities and they get lost.

While peppers are self fertile, they also can be pollenated by outside influence like pollinators and weather. If a grower wants to ensure the purity of that pepper, they isolate some or all of it.

I isolate by bagging a branch or cluster of buds before they start to bloom. Or I might grow one plant in isolation. I've seen local growers of produce isolate off a small section of plants out of their field. They cage in a few plants at the end of a row to save their seed for the next season.

Just saving the seeds from a diverse pepper patch won't guarantee you will grow a pepper plant true to the mother plant so not a good Idea if you care. Not also a good Idea if the plant is a hybrid to save any of the seeds because you won't get a true replica to the original plant.
You can isolate the plants you want to save seeds from and try to collect seeds but still doesn't mean the pepper flower wasn't pollinated by a flying pollinator moving across your garden or yard. Odds are low but still could happen.
The best solution for true seeds is to bag some flowers right before they bloom and once they bloom hand pollinate and re-bag until the blossoms fall off, and then tag them. Peppers are self pollinating so you shouldn't have much of a problem. Now I know for a fact Tomato flowers are only viable for a day or two, but not sure about peppers so hand pollinate as soon as they bloom.

I took seeds from a heirloom open pollinated Beaver dam plant packed in a patch with a dozen other types of pepper plants many years ago and grew them. they weren't anything like the original, the shape and size was right but flavor wasn't even close and rather nasty.

Thank you both. You have confirmed my previous beliefs. I know all about isolation techniques, etc. I was just surprised that the need for those techniques was not mentioned up front in these articles about saving seeds. I just read another one, and it did eventually discuss isolation, but not before I'm sure many readers would have long go stopped reading and assumed they could collect seeds from any old pepper. They had even discussed trading seeds with other gardeners, but had not yet mentioned cross pollination, or even hybrids.

I worry for folks involved in seed trades, who are hoping for reproduceable varieties. I will redouble my efforts to ensure any seeds I save are as pure as I can make them.

Martha