In Agile development, who owns the software “features” and how to you manage the development?

Question

Some development teams in my company are switching to Agile development practices and their developers' work seem to be diminishing to discuss and program minutia about trivial software features because of two week iteration cycles. And also because of "any developer can fix any bug" mentality. I have recently joined one of those team, transferring from another team in the same company...

I feel strongly that developers should own their software features from start (design) to finish (implementation and unit testing). Agile seems to be going counter to this thinking. Is there any truth to my perception or am I just living a bad implementation of Agile?

During our two week iterations, people somewhat arbitrarily get assigned new little features and bug fixes, depending on their workload during that cycle. Nobody seems to be owning responsibility of major features of the software. We spend stupid amount of times on trivial things, like adding a single button to a dialog during a two week iteration, complete with a story, daily scrums, code review, etc.

So in Agile projects, how does one manage larger features? Who own the responsibility: individual developers or the whole team? How does one extract him/herself from minutia and focus on longer term goals? Any feedback would be valuable.