Their intro video says you can run it on Cloudflare Workers, so I guess you can also run it locally using Cloudflare's local dev setup for Cloudflare Workers.

https://developers.cloudflare.com/workers/wrangler/commands/...

Their Contributing guide has a couple sections on getting it up and running: https://github.com/webstudio-is/webstudio/blob/main/docs/con...

Hi, we just launched Beta and didn't start yet working on documenting the self-hosting and self-publishing experience much.

It will come for sure. We are still focused on builder experience.

Good thing you can pay them to take care of those things if you don't want to invest effort in finding out for yourself.