It's going to be a slow burner, though, isn't it? The immediate effects are the myriad small inconveniences that are now in place that weren't there before (customs declarations, health insurance stuff, work permits, etc) plus the 'big ticket' EU immigration rules for people coming here or going there to live and work. (The end of free movement is the thing I resent more than everything else put together wrt brexit.)
So that's all shit, but for most people it's mostly small stuff (not for everyone of course - people with EU partners wishing to settle here, for instance, now face a mountain of paperwork and expense).
But the big test comes when one side decides to push against the limits of the agreement on alignment. That could come in six months' time or two years' time, or whenever, and it will be very apparent at that point what abstract notions of a theoretical national sovereignty count for in the real world.
And we all know which side it will be that will try to take the piss by attempting to undercut the other. As with most of this stuff, the result is that I'll probably be hoping for a strong response from the EU and for 'us' (ie the UK government) to be humiliated and forced to climb down. The UK's 'sovereignty' may have been increased in theory by brexit. In practice, it is likely to have been reduced.