Is it possible to take a train from Paris to either of the above, then travel between them, and return to Paris in one day WITHOUT a car? I believe there is a bus system between the two cities but do the connections make it possible to see both in one day? Many thanks.
Is certainly doable, from Gare du Nord in Paris to Compiègne is with a direct train not so difficult. The best option to my opinion is taking the bus to Pierrefonds for the final leg. For planning and buying train tickets you can use: https://en.oui.sncf/en/ and Rome2rio for bus routes. https://www.rome2rio.com/fr/s/Compiègne/Château-de-Pierrefonds
Nevertheless it’s worth considering a rental car. The area around Pierrefonds is nice with a bunch of cute villages and worth exploring if you ask me. Just have a look with Google Streetview in Vic-sur-Aisne around the château for instance, maybe a bit too far away but I liked to discover La Ferté Milon and the ruins there some years ago. Must be possible to put together an attractive itinerary.