I'm using Opera 35.0.2066.82 on my laptop, and Opera Beta on my Android phone. The problem appears to be with the desktop installation.
Briefly, I can't get sync to work properly because Opera won't stay signed in. When I start Opera, there is always a red X over the profile icon at the upper right. Clicking it brings up "Synchronization. Something went wrong. Synchronization has stopped because of an error. Please sign in again." Then I click the "sign in" button and go through the sign-in dialog, which tells me synchronization is enabled and shows me logged in. As soon as I do anything, however, that screen goes away and I'm back to where I started, with the red X. Clicking it again tells me something went wrong, etc.
I don't know if the problem has to do with sync, per se, or with staying signed in. For what it's worth, I don't seem to have any trouble staying signed in on the Android app.