Honestly, this is more of a usability concern than an accessibility one . . . though, the two often overlap. After all, if a user without disabilities has difficulty perceiving the success of an action, users with certain disabilities (visual, cognitive, reading, etc.) will likely experience that difficulty more intensely.
In the end, it's all about providing some kind of immediate feedback to the user, to let them know that their action did something (successfully or not) and whether or not that needs to be added explicitly is going to vary, depending on the functionality.
For some functionality (including the ones that you listed in your question) even though there might not be a change in the page itself, the user will receive immediate feedback, letting them know that the action was "successful":
- for "Download a file", the browser will initiate it's default download process
- for "Open a new tab", the browser will open a new tab
- for "Print the page", the print dialog will open
For situations like these, as long as you have clearly informed the user of what action will take place when the link, button, etc. is clicked, the very fact that the action occurs should be sufficient feedback to the user to let them know that the click was "successful".
The situations that may need additional feedback are ones without any immediate response. For example:
- for "Email me a copy", the user will eventually receive an email, which would confirm the success of the action, however, from an usability/accessibility point-of-view, informing the user that "An email has been sent", would be a more immediate indicator of the success of the action.
- for "Add to my reading list", the user should be able to navigate to their reading list to confirm that an item was indeed added, but the experience would be more usable/accessible if they were immediately provided with a message that "XYZ was added to your reading list".
TL;DR - For the sake of all users, if a page control does not inherently trigger an immediate action/reaction/response, provide your own feedback to the user, to let them know if their action was successful or not.