It can run in both the browser and/or the server. Because WebRTC unfortunately requires a STUN/ICE or signaling server, you still have to run or use a community server. A gun server is nothing fancy though, it is just another peer that helps replicate/synchronize data between browser peers and other server peers. Full WebRTC support is planned as an extension but isn't finished yet: https://github.com/PsychoLlama/gun-rtc .
See my other comment elsewhere in this thread about Pouch. Happy to answer any other questions you have! And please let us know what we can do to further clarify things in docs.