You can use Disqus to create a comments embed on your site and blog posts.
Though this does encourage your readers to sharecrop (on Disqus) their comments on your posts, it also provides them a single place to view the comments they've made across any/all sites that use Disqus for comments.
How to Export
(this section is a stub and needs help!)
How to export your comments from Disqus is actually two different questions:
- How to export comments written by you from Disqus (across all posts/sites you've commented with Disqus)
- How to export comments on your posts and site from Disqus (across all users who have commented on your stuff). Note: 2016-01-28 request for this
Exporting a blog's comments
Disqus provides the ability to export (XML) comment data from their system for people using it as a plugin commenting system -- that is comments written on your posts/site (across all users who have commented on your site). The export data they provide is meant as a simple back-up rather than as something useful that would allow the comments to be re-imported into another site. It appears that those who use it on WordPress can opt to have the Disqus comments mirrored separately into their database so if they quit using it they still own the data.
Exporting your own comments
As of 10/22/16 it still appears that there is no simple or direct method to export one's own comments.
No indieweb community members are currently using this on their own site(s).
IndieWeb community members that have tried this and their experience:
- Aaron Parecki: "Facebook tends to get more comments [than Disqus] for me probably because everyone has an account and is always logged in"
- Christian Weiske: "disqus doesn't work on my page"
- "I’m fed up with Disqus. It’s been useful, easy to plug on this blog, but it’s a mess for web performance, and I don’t own my data, so… bye bye. Webmention is now a great alternative, with more people implementing it, so let’s try to keep only that."
From cweiske on IRC: 
I'm using application/xhtml+xml as content type for my html pages, and disqus breaks because of this
contacted their support several times and they don't plan to support that
- 2014-10-04 https://twitter.com/semil/status/518546686524657664
[...] (also noticed Disqus disabled. have been thinking same. too much spam.)
Displays ads on site to public but not owner
Via https://twitter.com/fdevillamil/status/394368809597149184, in short:
I'm getting reports that disqus shows ads to everyone but the admins/site-owner and uses some very shady tactics to do it.
Supercookie used to do so:
Confirmed, looks like Disqus is using a super cookie to track me even after clearing cookies and then avoids showing the ads to me.
Compare screenshots made from different computers but from same IP address:
Screenshot of same spot with Disqus on a computer used to previously access Disqus (but not logged in):
Possible use of "browser fingerprinting" reported by former employee:
@zedshaw when I left they were trying to use browser fingerprinting despite my protests
The Disqus for WordPress plugin supports automatic syncing of comments between the Disqus system and the WordPress database. Which means there is no action required if Disqus is disabled for you to continue to have access to the comments people provided through it.
- 2017-09-06 Beware of Disqus! Disqus will continuously load garbage requests in the background of any page its on!
- Criticism: 2017-07-27 So long Disqus, hello Webmention
I’m fed up with Disqus. It’s been useful, easy to plug on this blog, but it’s a mess for web performance, and I don’t own my data, so… bye bye.[…]
Disqus accounted for three quarters of the page weight!
- 2018-07-16 Bye Disqus, hello Webmention!