Trading on insider information is illegal, but it's also hard to spot. I'd think if all the world brokerages got together and examined all trades within a couple of related connections to each Congressman, it would be illuminating. Congressmen may be under scrutiny. Their grandson's wife? Their sister-in-law's son? Probably not so much. Just look at the Trump family connections, new Trump Towers, all his crypto currency, and on and on.
Guess I should have specified blind trust or some such instead of just saying turning it over to a financial advisor.
As far as who is doing the trading - I suppose it depends on the Congressman. Some seem to not have much actual work product to show for their time in office, so they certainly have time to do it themselves if they wanted. After all, buying or selling a stock online takes well under a minute. And if you were going to do insider trading, I'd think the fewer people who were aware of it, the better.