None of the popular wrist-based trackers have an oximeter. You need red and infrared light for that, and it is really tricky to get accurate measurements at the wrist in any case. You don't measure respiration rate with oximetry in any case, oxygen saturation does not change that fast. The more advanced solutions are mostly based on Heart Rate Variability which shows some correlation to sleep stages. Source: also worked on this.

Can you shed light on whether there are a common set of algorithms that sleep trackers use or if they are generally all proprietary?

There are some common algorithms for sleep/wake classification from accelerometer input, but sleep stages algorithms are all proprietary as far as I know.

Which is another way of saying "no way to verify if they're bullshit or not".

Not true, anyone could take them to a sleep lab and compare to Polysomnograph-based sleep scoring. I'd love to see a large, independent study on this for a couple of the leading trackers.