Not any more. Some of the older versions did. In the latest builds SD shows in Bookmarks so that you can copy entries from SD on one machine to SD on another - but you may not want all the same speed dials at both work and home, for instance.
Do I really need to tell you this is not real sync?
http://blogs.opera.com/desktop/2015/03/opera-29-beta-speed-dial-tab-sync/#comment-1919168462
And that's why I created this topic:
https://forums.opera.com/topic/8987/sync-same-speed-dial-on-every-device