I think there are three ways to look at whether you should be reporting a meeting.
You can look at it in isolation. As a stand-alone concept, having both sides reporting the same meeting would make it easier to see if somebody didn't report it. That would be a plus.
Second, though, to come back to the circle you're trying to draw around all the recommendations on the table, if you put that recommendation alongside all of the others, such as “all oral communication should be reported; you don't have to be paid to register; it doesn't matter who initiates the communication”, you get the sense that you're looking at a major bureaucratic nightmare: MPs and lobbyists making records of and reporting every conversation they have about government with everyone they meet, anywhere, no matter what the context.
The third way to look at this is from a governance point of view. Charles is right. The commissioner's office would need to be greatly expanded. You'd be looking at an exponential increase in the number of reports coming in to that office. For every report coming in now, you'd have two. And if you put those other recommendations on the table, for every report coming in now you might have ten.
There was a well-received report last week on how to reduce red tape. I don't see how that recommendation supports this.