MastroLinux suggested that Arduino might set up a system similar to how the Language Reference pages now work with the content in a GitHub repository, where it can be edited by the community, then published to the Arduino website. If that is actually going to happen, then I don't think we need a 3rd party version of the Playground. However, I'm still very skeptical that will ever happen. It would be really helpful if we could get a firm official answer on whether that is something Arduino is going to commit to doing, and if so, what a realistic timeline is. It's been 1.5 years since Massimo Banzi announced that the official library reference content was moving to GitHub, but I haven't seen any work done towards that. It's hard for me to believe that the Playground project would be given higher priority (if any at all).
Probably the quickest path for starting a 3rd party Playground would be to set up the PmWiki software currently used by the Playground on a separate website:
http://pmwiki.org/
After that, there should be some way to easily import the contents of the Playground. It might be necessary to coordinate with someone from Arduino to get access to the raw content.
Based on my experience with the Playground, I'm actually not a big fan of PmWiki. The edit history is just terrible. I'm also not crazy about the markup. It would be much better if we could use Markdown, which anyone using GitHub, Stack Exchange, etc. will already be familiar with. I suppose it's possible that PmWiki has been much improved since whatever outdated version Arduino is using, or maybe there are modifications that can be made to improve it. However, converting the content from PmWiki to some other platform is likely to be a much bigger job (though perhaps not so bad if we found some magic converter program that does an acceptable job).
The big question is how this 3rd party Playground will be hosted. If some random community member grabs a domain and hosting plan and sets this up, what guarantee do we have that the bills will continue to be paid and necessary infrastructure maintenance work will be done? This is an issue because I don't want to do a lot of work contributing to a resource that's going to disappear after a couple years. That is why Arduino is the best organization to host this resource. I believe that Arduino does benefit from community-generated documentation.
I have actually given the idea of starting a 3rd party version of the Playground quite a bit of thought since this announcement was made. Hosting it on GitHub seems like a good option. Public GitHub repositories are free so there is no concern of paying for hosting. If the owners did end up abandoning the project without passing on the torch, it's simple enough for someone to fork the repo. It is unfortunate that a GitHub account is required to contribute, but a significant number of the potential contributors already have one and it's quite easy for people who don't to set one up. I've noticed an amazing lack of spam/abuse/vandalism on GitHub. I considered a GitHub Wiki, but that feature is surprisingly very poorly done. In the end, I think a standard GitHub repository would be better. The potential issue with that option is that it MUST be actively administrated. Valid PRs must be merged quickly, otherwise people will not be motivated to make the effort to contribute. It's a shame you can't create GitHub repos that give everyone push access (as is the case with public GitHub wikis). I think the key would be for the repository to have a lot of administrators with a liberal policy about accepting PRs. The big problem is the work of transferring the Playground content to the GitHub repository. I know there are some tools that will convert the HTML output of the Playground to Markdown (perhaps pandoc). I also see there is a PmWiki "cookbook" that allows PmWiki pages to be shown as Markup output: