web site session sites?

I sometimes find myself sitting at someone else’s computing using their web browser. With all of their sessions and cookies. So say, going to a given website I get their login.

I can log them out of their accounts and log me in, but thats annoying. Some firefox extensions exist that seem to support multiple users per browser, but that requires extensions.

I wonder if it would be possible to setup a proxy website that would store all my various session info so I could retain it as I switched machines.

I’m not exactly well versed in web stuff these days, but it seems like it should be doable. Maybe such a thing already exists.