I've witnessed both passengers not getting off when intended as well as those not getting back on in time at a 'longer' stop.
In coach on LD trains, the conductor & assistants will place a seat check indicating the destination station with a standard 3 letter code for each occupied seat. On the NEC, blank seat checks may or may not have a corner torn off, partially folded, or placed at an angle or horizontal to indicate destination with vertical being the 'last stop'. Regardless, in addition to PA announcements, one of the train crew members will usually advise each passenger that their station is coming up in 5 minutes or whenever. It's up to the sleeping car attendants to ensure their passengers get off as intended.
Maybe 40 years ago already, I was on the eastbound Broadway Ltd having just left Harrisburg and walking from the diner back to my roomette when a seated passenger started yelling he had to get off now as he slept through Harrisburg. The conductor told him he'd have to ride to the next station and make his own way back to Harrisburg. These days, the app on the conductors' smartphones indicates how many should get off and on at each stop. If it's a small number (under 20 or so?), he'd keep count and know that someone that is on the train failed to get off at that stop. I've heard announcements to the effect 2 or 3 times in the past 8-10 years.
Passengers getting too far away from the train at an intermediate stop are always prone to missing it. That's especially true for 'smoke stops' that are scheduled for about 15 minutes but if the train is running late, it's shorter. These days, the pre arrival announcements always indicate to stay near the train. But it doesn't always happen. Maybe 15 years ago, a passenger didn't get back on after a crew change/refuel/watering stop at Havre MT. I heard on my scanner about 5 minutes after we departed that so-and-so was still at the depot and that the crew should take his belongings and leave them at the next station stop. What Mr X did after that, I have no idea.
In short, if someone misses their stop, too bad. Amtrak can't put them off in the middle of nowhere so they have no choice but to take them to the next station. Although, as an aside, I have witnessed multiple occurances of the train stopping at a grade crossing with local police removing one or more unruly passengers.