If it has access to the world wide web, then you can surf the web globally and use global applications on it. It's not a local web. It's a decentralized web peer community connected to the world wide web.
If your network is connected to "the internet", your traffic can again be censored by ISP's and their routers.
You'd need to have a
local global network without any ISP. And this is not the case with your mesh network
connected to the world wide web.
You can also encrypt the signal between peers so that they are anonymous to each other.
Encryption does not make you anonymous. It protects the content against tampering and curious participants of the network.