Namespaces provide the first and most straightforward form of isolation: processes running within a container cannot see, and even less affect, processes running in another container, or in the host system.
Each container also gets its own network stack, meaning that a container doesn’t get privileged access to the sockets or interfaces of another container. Of course, if the host system is setup accordingly, containers can interact with each other through their respective network interfaces — just like they can interact with external hosts. When you specify public ports for your containers or use links then IP traffic is allowed between containers. They can ping each other, send/receive UDP packets, and establish TCP connections, but that can be restricted if necessary. From a network architecture point of view, all containers on a given Docker host are sitting on bridge interfaces. This means that they are just like physical machines connected through a common Ethernet switch; no more, no less.
How mature is the code providing kernel namespaces and private networking? Kernel namespaces were introduced between kernel version 2.6.15 and 2.6.26. This means that since July 2008 (date of the 2.6.26 release, now 5 years ago), namespace code has been exercised and scrutinized on a large number of production systems. And there is more: the design and inspiration for the namespaces code are even older. Namespaces are actually an effort to reimplement the features of OpenVZ in such a way that they could be merged within the mainstream kernel. And OpenVZ was initially released in 2005, so both the design and the implementation are pretty mature.