I came into reproducible research in the context of air pollution and
health epidemiology, where some of the research can be controversial
in that it can affect national environmental policies. As a result,
some of my work and the work of my colleagues has been challenged by
industry groups. Scott Zeger and Francesca Dominici felt that the best
way to deal with these (time-consuming) challenges was to make all of
the data and code available so that people could conduct their own
analyses on the same data that we had. If their approach was truly
better, they should be able to demonstrate it on our data. 

My feeling was that this was the right thing to do on paper, however
it did not account for the possibility that some users of the data
would not hold themselves to the same standards of scientific conduct
that we might. Therefore, the release of the data, in my view,
encourage the misuse and poor analysis of the data, obfuscating the
issues involved. While it is always possible to have an honest debate
and to refute a faulty analysis, often this debate is lost once the
headline in the New York Times has been printed. In the end, I think
we as scientists do have an obligation to disseminate our work in the
most transparent way as possible, but this needs to be balanced
against the potentially nefarious interests of some parties.

Regarding the questions posed:

Investigators: I think an investigator's responsibility is relatively
simple: he/she should be able to reproduce his/her analysis upon
request at some point in the future. I think this is a minimum
standard. One way to imagine this would be to consider a hypothetical
audit of a given project in the future. Could an investigator
reproduce a published result in that setting? Making research
reproducible by others is a much more complex problem and depends
critically on the resources of the others.

Journals: I think journals should facilitate/coordinate the
publication of computational research. They should not be responsible
for somehow validating that work (as they are not responsible for
validating other research) but to make sure that the work adheres to
some minimal standard. In the computational arena, one possibility is
to employ a technical/computational editor as is done at
Biostatistics, Biometrical Journal, and Journal of Statistical
Software.

Institutions: I think the primary responsibility of academic
institutions is to teach students how to conduct reproducible
research. We should teach them the skills, tools, and best practices
so that they know what is a "reproducible habit" as opposed to a "bad
habit". The set of skills that encourage reproducibility should be
considered core skills, much like knowledge of a programming language
or calculus are core skills.

Funding Agencies/Regulators: Funding agencies should play a role in
developing the infrastructure needed for supporting reproducible
research. In particular, long-lived repositories are needed for
hosting data and code so that it can be available to others long after
a paper is published. Also, funding agencies could encourage/fund the
development of tools/software that allow investigators to easily make
their work reproducible by others.