"Professional juries" is essentially an oxymoron. The basic idea of the jury system is that people who judge your guilt are your fellow citizens, not government functionaries. (Whether this is good or bad by whatever metric is beside the point.)

Besides, in common law jurisdictions, you can typically waive your right to be tried by a jury and have a bench trial, where the judge is responsible for findings of fact as well as law. So basically, you already have the option to be tried by a "professional juror."

Not prepared to advocate professional juries, but off the top of my head, I'd have a professional juror train in law, statistics, demographics, forensic science, and cognitive biases.

Even better would be subsidized prediction markets. That way people train themselves as necessary to get results.

There'd be some kinks to work out, since you don't always get the answer handed to you after all the bets are in, but I think this would be a solvable problem. You could find ways of having occasional payouts based on cases where there is slam dunk evidence which is withheld and the professional bettors don't know if their decision will affect the defendant's sentence or just their pay. You could also try rewarding consistency between separate prediction markets in the hope that "our actual best guess" is the most salient Schelling point.

I'd be curious to know whether advocacy of professional juries was common on LW.

A great idea, except for the corruption magnet. (But probably no worse than judges as they stand.)