The canonical link is intended to give a hint to Google as to which page is preferred for their index in the case when you might have duplicate or highly similar pages on your site. The cross site canonical is for use when you may have more than one domain and both have duplicate or closely similar content. The canonical tag may or may not pass any authority, it is meant to signal which of the two duplicate/similar pages should be indexed. Google may ignore canonical links that are misused.

A recent discussion might explain this better: [webmasterworld.com...] although that is not the same scenario described here.

To quote NickMNS in that thread -

A Canonical link signals to search engines, that the two pages may not be exact copies, but are essentially the same, and thus when someone is looking for either of these pages please show the page in the link instead of the other one.

@not2easy I bet you can't guess who liked your post!

@mancunian to put in terms of your questions:

I am concerned that the page on site A may no longer show ...

You are right to be concerned, because by adding the canonical link you are telling Google don't show page on site A in the SERPS, instead show page from site B, they are basically the same and I prefer B.

Did you check to see which page ranked highest in google for their main keywords before you started adding links and canonical tags? Then you could do a "before and after" comparison of the google traffic to each page separately and also the combined total for both pages.

Edit: Additional Note: Actually it's the total google traffic to both SITES, not just those two pages, that matters. Because if you strengthen one page on a site, it might also help the rankings of some of the other pages on the same site.

Great answers. Thank you appreciated.