The fMRI Data Center provides access to “peer-reviewed fMRI studies and their underlying data.” The center envisions that the data will be interoperable with a federation of other data sets.
Data sets are submitted by authors, and the center does all conversions. The center also ships data sets to requestors in the format they request. The FAQ outlines the types of data that they ask authors and others to submit (raw vs. processed).
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The website linked above is no longer active. Please see OpenfMRI for a possible source of shared fMRI data: https://openfmri.org/
Also see this 2013 article in PubMedCentral:
Toward open sharing of task-based fMRI data: the OpenfMRI project –
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3703526/