EL,
Most scanners are larger than a Walkman and therefore would not fit easily into your pocket. A few of the higher priced ones are indeed walkman sized, however you'd still have a problem fitting it into your pocket since the antenna would be sticking out.
All scanners do come with belt clips so you don't have to carry them by hand. I usually travel with my scanner under my shirt, so you can't really tell if I have a scanner or a Walkman. In fact since I use a very small earpiece, you might not even know that I had a scanner. I've never had any trouble sitting on a train with my scanner clipped to my belt, even on the NYC subways.
As for disturbing passengers, since you must wear headphones in coach, you won't be bothering passengers. If you are in a private sleeper then you can remove the headphones assuming that you don't turn the volume up too loud.
You can check out
this page from Radio Shack to get a feel for price and size. Scanners are also available from a few other sources including Bearcat.
You might also wish to check out
this page from Prodigy's On Track On Line. They cover the basics of how and why a scanner works, additionally they list all the frequencies assigned to RR's. Finally they give you a comprehensive list by route of the frequencies used by Amtrak.
If you have more questions, just let us know.