Unfortunately the loudmouths will remember this come review time, and obtain their vengeance.
Flat companies seem to drive people to do highly-visible hero work to ensure they can garner enough positive reviews, rather than the quiet but essential work that leads to 'dunno what he did all year, requires improvement'
You didn't want to be either one of these people when the layoffs hit. Your own manager might go to bat for you, but nothing says "keeper" to other managers like a nice string of public and visible wins.
If nobody knows what you are working on, that's a problem in any company. It's probably worse in a flat org, but it's still bad in a tree.
Self promotion is necessary, at least to some extent. You could be a great engineer but in a tree structure you're going to get passed over for interesting projects because nobody knows what you can really do. In a flat org you'll ideally get feedback that you can work on before your ass gets fired, but you can't count on it.
Contrast with a traditional company where the loudmouth is a manager, you're not allowed to move your desk, and dissent (like working on something else, or ignoring the guy) leads to HR action.