What is the process, I ask you good C.SE'ers?
Step 1. Ask on meta.
First, no matter how much discussion might have happened in chat or comments or wherever else, step 1 for getting actual consensus should be a meta post.
Step 2. Read the pulse.
Once the issue at hand is spelled out, you listen to answers and votes. These are the pulse you're trying to read. The scenario could play out a lot of ways and in the end it usually comes down to a judgement call on the part of moderators to call something consensus or not.
- Perhaps the issue was simple and extreme levels of up or downvotes on the meta question indicate the community stance.
- Perhaps a couple clearly different answers come in and extreme voting one way or another makes it clear which one to go with.
- Perhaps nobody gives a flip, you get a couple of votes and the whole thing fizzles.
- Perhaps it generates a lot of nuanced answers and voting is not clearly going one way or another. In this case let it run for a long time prodding for process as necessary. Later a new post trying to get less discussion and judge the approval of a susinct version of events might be in order.
Step 3. Rinse, wash, repeat.
This judgement call can easily be challenged by the community, usually with another meta post asking whether something or other should be considered consensus. You'll notice a lot of tricky questions in the past often involve several questions that are basically discussion starters and then finally a later one that spells out the conclusions.
Sometimes the same issue may need to be aired out all over again if there start to be rumblings of change in the community.
Thankfully C.SE has pretty good meta vote participation so it really hasn't been that hard to call things here. It's a lot harder on BH where meta votes might be +5/-2 and you have to call that consensus. Here once an idea gets clearly expressed after concerns have been discussed, we're more likely to get +9/-1 or even +23/-0. Either way it's a pretty easy call.
Edit: As I see it, it's perfectly OK for a non-mod to go through this process. The posting, answering, voting and even "calling it" one way or another can easily be done by anybody with enough experience with the community to pull it off. As long as they have a good feel for what normal meta participation is and what indicates good feedback vs. sketchy feedback, I don't have a problem with non-mods doing this. The only a time a mod would be specifically required is if there were a dispute over the call. Person A asks about their pet issue, gets a couple of votes, then goes around commenting about how their issue has community consensus. Person B objects and says that there was another answer on that issue that was better received by the community. At that point a mod should probably be the one t make a call and say yes or no about whether something can officially be called "consensus".