I've been spending some of my free time working with two open-source projects that are building new, community-driven Q&A platforms. (Yes, two. We're cooperating, including on common interfaces, but have some different goals. We didn't know about each other right off.) I don't have useful programming skills to contribute, but I'm helping with other aspects, including functional design, some feature design, and general cat-herding (on the larger one). Also, one of them asked me to serve as doc lead. :-)
Codidact is a platform for networks of sites on specific topics, much like Stack Exchange is a network of sites. Lots of (current and former) moderators and users from Stack Exchange are involved. (No I did not start this project; I was recruited after it had started.) We're talking about better management of comments/discussion/feedback, and about answer scoring that takes controversy into account, and tying user privileges not to a single "reputation" number but to related activity on the site. We're also talking about allowing more per-site customization, the trick there being to support customization while preserving the sense of an overall network. We have a wiki, a draft functional spec, a front-end design framework, and a forum where we're hashing out a lot of the details. I hope we'll see a database schema soon.
As you can infer from all that, we don't have running code yet. However, we have one community that has been pretty much destroyed on its previous platform, and the Codidact team lead had previously built a prototype Q&A platform, so Writing has a temporary site now, as a stopgap and to keep the community together, while waiting for Codidact to be ready. (Site introduction.)
The team building the Codidact platform will also run an instance (a network of sites). Others are free to take the software and run their own instances if they want to follow different policies or prefer to have full control.
TopAnswers is being built by a few people from the DBA site on SE. They are being much more agile than Codidact is; they have a running site already, which gets improvements on a near-daily basis. Chat is tightly integrated; they actually built chat first so they'd have a place to coordinate building Q&A. They have an interesting voting model where people who've gained more stars (reputation-equivalent) can cast multiple votes on a post, essentially giving experts (to the extent that stars = expertise) optional weighted votes. They also integrate both meta posts and blog posts into a site's main question list instead of isolating those types of content elsewhere. I find this idea intriguing and am advocating it for Codidact too. (The link I provided is to the network-wide meta site. If you choose "Databases" from the selector at the top, you'll see what a "regular" site would look like.)
When some sort of incorporation is needed, both projects are planning on going the non-profit route (a la WikiMedia), so that the communities, not profit-seeking, remain central. Right now I think both are running on donated hosting.
Both approaches look interesting to me, and I can see some communities preferring one over the other. I'll be interested in seeing how things work out -- what ends up getting implemented on each, what lessons both positive and negative we learn from past experience, what changes stick, and where individual communities end up being active.