ASM - Managed by Exin. You DO need to attend a training course by a registered provider and show the certificate. The training can be done online rather than in person, and the exam fee IS included in the training course. I did mine with Simplilearn and I believe I negotiated the price down to $599 USD. I was a little disappointed with the course content, especially after sitting the exam and seeing several topics that were not covered in the training. My strong recommendation is to read the Scrum Guide, and Mike Cohn's two books "Agile Estimating and Planning" and "Succeeding with Agile" as I saw many exam questions directly from all these books, like over 50% of the exam. If you fail the exam, the re-exam fee will be $250. I studied a few weeks for this one as the topics were so broad compared to the regular Scum-only content provided in other certifications.
PSM - Managed by Scrum.org. You do NOT need to attend a training course by a registered provider, although they highly recommend it.