Blogger authentication cookies, which are created under "Blogger.com" and read under "BlogSpot.com" - or whatever non BlogSpot address the blog in question is published to - are affected by the "third party cookie" option. "Third party cookies", as with all cookies, are affected by filters which may be found in any section of a layered security policy.
This option is most commonly found in a browser filter setting. It may also appear as a cookie or script filter, in some browser security add-ons, in any anti-virus, firewall, or "security suite" that runs on the computer, and in a network appliance which may be labeled as a "modem" or "router".
CAPTCHA ("word verification") screening does not, by its basic nature, require access to "third party cookies". Blogger commenting options, however, make access to the Blogger authentication cookies necessary.
The nature of the comment posting form is an additional issue. Only blogs which use an inline comment form ("Embedded below post") are affected by "third party cookie" filters. Both "Full page" and "Pop-up window" forms run under "Blogger.com", and are not affected. If you don't believe me, examine the URL in the browser address window, with any comment form displayed.
A blog that uses an inline comment form is affected by two comment settings.
- "Who Can Comment?" allows blog readers, other than members, to post comments.
- "Show word verification for comments?", when selected by the blog owner, requires only non blog members to solve a CAPTCHA when posting a comment.
If you can't comment on other people's blogs, or if various readers of your blog can't comment on your blog, and if the blogs involved use an inline comment form, chances are very good that "third party cookie" filters are part of the problem. This isn't an attempt by Blogger to hijack your browser, it's just the way authentication, and comment posting, work. And it's not a problem that Blogger can correct, by themselves.